diff --git a/math_linear_algebra.ipynb b/math_linear_algebra.ipynb index 3be5180..817c930 100644 --- a/math_linear_algebra.ipynb +++ b/math_linear_algebra.ipynb @@ -2,7 +2,10 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "**Math - Linear Algebra**\n", "\n", @@ -13,7 +16,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Before we start, let's ensure that this notebook works well in both Python 2 and 3:" ] @@ -22,7 +28,9 @@ "cell_type": "code", "execution_count": 1, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [], "source": [ @@ -31,7 +39,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "# Vectors\n", "## Definition\n", @@ -73,7 +84,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Vectors in python\n", "In python, a vector can be represented in many ways, the simplest being a regular python list of numbers:" @@ -83,7 +97,9 @@ "cell_type": "code", "execution_count": 2, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -103,7 +119,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Since we plan to do quite a lot of scientific calculations, it is much better to use NumPy's `ndarray`, which provides a lot of convenient and optimized implementations of essential mathematical operations on vectors (for more details about NumPy, check out the [NumPy tutorial](tools_numpy.ipynb)). For example:" ] @@ -112,7 +131,9 @@ "cell_type": "code", "execution_count": 3, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -134,7 +155,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "The size of a vector can be obtained using the `size` attribute:" ] @@ -143,7 +167,9 @@ "cell_type": "code", "execution_count": 4, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -163,7 +189,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "The $i^{th}$ element (also called *entry* or *item*) of a vector $\\textbf{v}$ is noted $\\textbf{v}_i$.\n", "\n", @@ -174,7 +203,9 @@ "cell_type": "code", "execution_count": 5, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -194,7 +225,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Plotting vectors\n", "To plot vectors we will use matplotlib, so let's start by importing it (for details about matplotlib, check the [matplotlib tutorial](tools_matplotlib.ipynb)):" @@ -204,7 +238,9 @@ "cell_type": "code", "execution_count": 6, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [], "source": [ @@ -214,7 +250,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### 2D vectors\n", "Let's create a couple very simple 2D vectors to plot:" @@ -224,7 +263,9 @@ "cell_type": "code", "execution_count": 7, "metadata": { - "collapsed": true + "collapsed": true, + "deletable": true, + "editable": true }, "outputs": [], "source": [ @@ -234,7 +275,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "These vectors each have 2 elements, so they can easily be represented graphically on a 2D graph, for example as points:" ] @@ -243,14 +287,16 @@ "cell_type": "code", "execution_count": 8, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -267,7 +313,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Vectors can also be represented as arrows. Let's create a small convenience function to draw nice arrows:" ] @@ -276,7 +325,9 @@ "cell_type": "code", "execution_count": 9, "metadata": { - "collapsed": true + "collapsed": true, + "deletable": true, + "editable": true }, "outputs": [], "source": [ @@ -288,7 +339,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Now let's draw the vectors **u** and **v** as arrows:" ] @@ -297,14 +351,16 @@ "cell_type": "code", "execution_count": 10, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -321,7 +377,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### 3D vectors\n", "Plotting 3D vectors is also relatively straightforward. First let's create two 3D vectors:" @@ -331,7 +390,9 @@ "cell_type": "code", "execution_count": 11, "metadata": { - "collapsed": true + "collapsed": true, + "deletable": true, + "editable": true }, "outputs": [], "source": [ @@ -341,7 +402,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Now let's plot them using matplotlib's `Axes3D`:" ] @@ -350,14 +414,16 @@ "cell_type": "code", "execution_count": 12, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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BFhHTKiESiTgmT7jdr4CFkfJfvW11tDab9ELCDigvzdB9Zt0jxWIR9913H/bu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\nkybhzDPPRDqdRi6XQ1dXlypNVEMrdWsXByOyBNn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JE273K2BhpPxXb1sdrc0mvZCwA8pLM3SfWfdIsVjEww8/jL1796KzsxM7duzA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\n", "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -376,7 +442,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "It is a bit hard to visualize exactly where in space these two points are, so let's add vertical lines. We'll create a small convenience function to plot a list of 3d vectors with vertical lines attached:" ] @@ -385,14 +454,16 @@ "cell_type": "code", "execution_count": 13, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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GIWlGy09tJCp2otmNkyhb0u3s7FQSTVY7c6nJFEbbROolHyJyvjk5LbnNgpb9\nyWTSNq8t+9Ix0svBLNQeUD6JRQ+pLMuKxFGu8gQAxcrGQqv8lydhknHKEfn81FrSDHvekiQpL7HW\n1lZHGpg7hbIl3UAggLq6OsMJLDUQ6ahVetnVZlGWZYVsq6qqsH79eowaNQojR460NC77gvD7/bb0\nXWDnamcvBzNQI2JKlIZCIcfkCbeSXGooJMdIkva2OmqbTZZDwg7IL82w5d70XWezWfz0pz/FwYMH\n0drait27d+Pss8+2/MyeddZZyj0fDAaxefNmS+PxKFvSpW3HrXpsCWR74iu99EJrHiyRs2Nffvnl\n3byTRo3o7LgkI1iRV/JVvBX6rNuwIk+Um3sC0NedS2uzSYqSyxVq555IJBAMBjF+/Hh8/PHH2L17\nNy677DL885//xDPPPIOZM2eaPp7P58O6devQo0cPO6bfDWVLugQrDz3rWxUEwVJky8+DJbBAINDN\nw5vPp1toztStjPZlCwQCStMaM6CHt7W1FT6fufaNVl4gdkGvPFFJ7gk9Vjb6Q70XnEpSurk6ICK+\n6KKL0NbWhlGjRmHp0qVobW3VLCDSC7p+TuG0JF2+SCAcDkMURVt0YZYU8xEY79PVM2e91WlGQGPK\nsoxYLGbLmKUEsx5bAt80xk44RVL8KoDcEYFAwFD/iVIGe+3YRJodCTVBEDBz5kz4/X7ccMMNuP76\n6y2PyaJsSddM9pwnRCoSoGYgViGKItra2gBo958lvPHGG/D5fJg6dWrB81CzapmNlNm5so3aOzo6\nLBEuEVipP6yEQvIEbaxJ170S5AktrdiurZTcjHTZY7W1taF///62jf3mm2+iX79+OHLkCGbOnIlR\no0Zh2rRpto1ftqRL0EM2haJEq8thtgyWxi50802bNq3gkpwlRrsKJqjJDVtNR+RiFtRPGMht75jJ\nZMqKoNiomAi4qqqq7OWJfGSoZmXjk3al1n+Cf07s3qqnX79+AIDevXvjiiuuwObNmz3SBXIjXS39\nhciW3XacNlHkxzJDumwBQjAYhCRJuk3a+W5UtTLjQjd2oXNgXQ5qjgQzEk06nVaq3WKxmPJgplKp\nirE7mZUnyrW4AzC/lZIkdW1K6VbEqyYvWAU9z7FYDIlEAq+88gruuOMOW8YmlC3pEigqYSHL+nra\nEoySrtrGjxTt6gXdxPQw0xwSiYSpvrY0ptq/sTY4O+xfbGkx9SP2+/050ZAgCEoj6UJ2p3IkKDvc\nE25auayfwLB3AAAgAElEQVTCyMsnk8mYKnLQC/662RnpHj58GFdccQUEQYAoirj66qsxa9YsW8Ym\nlC3pakW6bMMYvUtyvaTL915gI1CjxP3+++9j7969+MIXvtBteW+GGPlz1HI55EMhElCrTKOIh58L\n3yvYSGNxlqDUfKduwGyVnRH3BJ0TlYA7uVR3clz6bkVRRDAYVKLdfEUOVhri8N+Nnb10hwwZgh07\ndtgylhbKlnQJRHZGeg1oQYt0nOi9MGrUKIwaNQrJZFLxxQJdTWnMPCDs8fVsKMl/Nh/0VqYZmbce\nglLznRJxu2FJs4Oo8kWI6XS6m+G/nKN/Vs+n83BqKyX2Z/F43CsDdhMUJYmiiHA4bKrXgNbvGykH\nNuuiYKPQEydOWFpyyrKMtrY2w20h841H0oQblWmFlrCs51SWc8uAy0knBk6dq9/vz5FhnCjucNNV\nkA9GVzxaVjb+fLLZrGOboDqB8pmpCuLxuKIf1dXVWbqx2C+T10HtKgdmE3uCICh6MD8HoyDZQ5Ik\nSy8e9vyNVqY5Cf5hpfOlMuBK0omNyBOl5p4wQ+5aL9p8/ScIR48eNd18Px8kSUJTUxMGDhyI5557\nzvbxy5Z0BaGrX2soFFK2Abc6Hj28ZsimEGGyXttIJILjx4/jj3/8I1paWkzPmV32k5xiZcsSNgIX\nBEGXDsyet5tZ63LUic3ADvdEqUS6RqDlKWYtbOvWrcPSpUshiiIuvfRSTJgwAf/+7/+OCy+80NKx\nV69ejdGjRyuee7tRvgXZ6Oq/oOZeMAp6WNvb201v/KhFuqIoKptVhkIhZbPKhoaGboSrN9KlxFtr\nayuAUxtKWgVtyhiJRCyXAhcDRDjsBpTRaDRHaiHvM51rMplUHmI3dGK7QC+dYDCotAatrq5GJBJB\nIBDoZpekBkZOnqvT5E7nTFrx3LlzsX//fowZMwbf/OY3UVtbi08++cTSMQ4ePIgXX3wRCxcutGnW\n3VG2kS5wKgowWyfNFk1QBGqWvOhmoxuvkNdWKxGV72HgtWA2Ejd7DWiesiwjEAiY3g2YP7dSITC9\nOjHvJ6b/d5JI7B5b7VypGCYYDCqrACfkCTe/b/a6dXR0oL6+HrNnz8bs2bMtj33zzTfjvvvuUwIa\nJ1DWpAt0Jzs94KOAaDSq9GGwOhfa0pqcDvX19XkTdXoSU7wWrLbHmVGi44slBEGwpAUTcZFdqNSh\nJU/wWmIikSh7nRhANweLU8Udbl8TO3vpvvDCC2hoaMDEiROxbt06x+7jiiBdtYymFrQ2frSjiTgA\ntLe36ypCEEURP/7xj3HLLbfknAs/B7LCUSReyJFQ6DrwSTKaJ/UaMArSTNvb2+Hz+ZSHFzi1DU0p\nJHn0gCViIiA2QqwUnRiwvzWmm7oxW1RkZ2HEm2++ieeeew4vvvgiOjs70d7ejvnz5+OJJ56wZXxC\nWZMufcl6orxCfQzMLolZEpNlGTU1Nboax/j9fixbtqzb+dAc2Kq3aDRa0Hdc6IZnpRSfz1z7Rn48\nOm+ga/cO2m6IGghRlRobRakRVSlCzXPK/iyfn7iUOnYZIUMr7gk3JSX2nOwsAV65ciVWrlwJAFi/\nfj3uv/9+2wkXKHPSJeT7wvX2MTB606h5bWkpqnfOapAkKacU2I4tcthomQhcbT56k3j8du7xeDzn\nQaWHNxgM5kRRekuBnfQCG4HWdTerE/MvGzZiK1XodU9QL+dEIuG4FMPep3ZWo7mFsibdfJEubemi\nt4+BUdJhm+gQsZjRVVl5hKIIs4UIvMzCvnD0RMuFoEbeFAnpmZseixffzcruun0noUcnZl829HM6\n33I4RwIvT9DzFolEXNm5gz7b1tbmSDXajBkzMGPGDNvHBcqcdAks2ekp2dUao5ADQEsPVpuHHqxZ\nswZf+cpXEIlElCRZKBRCdXW17jH4c6CHnC3b1ds4R2vufDvIQgk3vdeh0HKWr9tnCZh+r9RJKt/L\nJplMAkDOduxqEbGVc3TrGvFyA/vvheQJo30YeHmhb9++jp2XE6gY0iViMLKDLz9GPolCawtyvWPw\nkGUZ1157rUKONTU13RrHGAXprGaugdr5kB+YHA75nBh2gSViVnPmXQWA+lK22BqqHrDnSJFiOenE\nRqBXnjDSPJ0l3ba2NgwfPtzVc7KKsiZdfllupWRXS6LQ6iqmBT2kq9WQxqyDgPRlWe5q/GO1bJfG\nI3+nnvGcTqKwERRF8tFotKCGWsrOCZY8rOjE+c7RzUjXyHGMuCeA3BUAezxP03UZoiji5MmTEISu\nkuBYLGZ6LF6iyNfwO98Y+cBHzHxDdT0SBw+WwH0+H6LRqKWXDo0nCPrLgIuFQhpquTonWOjRialp\nvFpSshw80wS9ThGgq+/C3LlzUV9fj+rqavh8PowfPx41NTWmj59KpTB9+nSk02mk02nMmTNHcTPY\nCaHAl1LS3xhFPKIoIpvNmtZCgS4Cj8fjCIVCSqObSCRiSKIgsuKr2niNNRwOQxAE/P73v8d5552H\nYcOGAYBSpqnn5aEmebS3tyteXqNob29XtFOjHcqOHz+O+vp6pfkQFYhEo1HD89ADup56v2+1oodC\nzgkqlnFyo07yStvRIYuPElkZBugqjnDST0wrLbaBkxOg776qqgpbtmzBAw88gJ49e2Lv3r04fvw4\nPvjgA0vjd3R0IBqNIpvNYurUqbj//vuVfQwNQvMCl3WkKwiCssOp1cIGyipns1nbJIpCEfOXv/xl\nw8fQ29vW6HiZTAbBYNAWi1qpwYxzglYc5VL0oBUlUttTQRDy6sQnT/qwf3/X9Rk2TELPnsaO73ZE\nHQwGcf755+O+++7DI488gpqaGlvmQIECrR569OhheUweZU26BLPGbNZrSzerVYmCokUzrRHznQdl\nu/Ntu2M0kceOV1VVpbp/nF6k02mkUilFW6TrUKpkVWgpSw+d2m7AdhGx09eHPUfWm82+bLLZLA4f\nFvH881Xw+VKQpGPYsaMPLr/cj169jJ1jMbRj0vbtOr4kSWhsbMS+fftw4403YvTo0ZbH5FHWpJvP\np5sPal5baiJuFaIoorW1VdcWOWySAFA/DzNJrXxgdVu2aU4ikTA1HmlsyWRSOVfWXVBOpbIsSVHk\nHwgEcqSJcnMVqBE7rRAJBw74EA5nkEj8HqnUMfh8Z2D79itwwQWnOrcVKnZwq9BD7XzsPK7P58P2\n7dvR1taGWbNmYf369bb7dcuadIHc3gt6wG7rw2qX9Hmz0Ucmk0EymYQsy4jFYrq0wPXr1yMWi6G5\nuVn151Z626qBLW6wug09690Ful5cREbUuyAcDiuRY7mRFQu1KjmrroJSgiQBsnwUqdRRADIk6Riy\n2TiqqxsKFq/QeRajBNjJY9bW1uI//uM/sGXLFo901aCHMNR28OWdAzSOkYeE34Y9m83qTr7827/9\nm+p5aL0YCkHrOthZmcbKEqRTt7a2IplMKmMSEdMDKgiCksjhix/KlYgryTkxfLiM99/vBZ+vBpIU\nh893BsaNO0O3m4C+Q/o3t79Du45x9OhRpQlUZ2cn/vKXv9i+/TpQAaRbKNI1kngyEu3xPR1CoZDS\nVMcsKKogF4KZVoss2EReocq0QufOyxI1NTWKUyESiUAUxZwOYxT9sEtRNptOS1z2BaUVNVrZOdZN\naHlPtcqA6QXLVmfZDT1BREODjC98IYBdu65FMnkU5557BgYOVA8ctPzEdF/QPeFU5M+ejyiKtm4j\n9emnn+LrX/+68p1dc801uOiii2wbn1D2pAucIgz2CzHjtdVDuvy4bJWWGW2Zbkyq/ALMbcHOHp9P\n5FndUJLvuUDyAetrJg2UXhTsJpK0PFUjT9aXTETMPpRqREznmc1my4qICfQd0TUthZ4TffvK6Ns3\nAMB8SW0gENAd+evRidXAlwDb2Xdh3Lhx2LZtm23jaaFiSBc4pfEUyvLnG8ct9wAA7Nq1CwcPHsQF\nF1yAYDCImpoapS+tGdBN3tbWplsHZufOF2bwPReqqqqUhwg41VTI5/Ohuro654ELBALdtt9WI2L2\n4WOPy86LJ2LyZZPDoNx0VCIYQegq6lGTXezq22skwfXBBx+gV69ettmktCJ/vTpxofN0qtmN0yh7\n0mWjTIpAzWb587kH1LbI0fN5NVB0M3jwYAwZMgTRaFTZ18psckAURaWMmJJkduq29KDQOSaTSSVZ\npsdqRuRphIhZ6UjturBG/HLTUXmwy3atnhNOJyNTqVSOBGQEenMhRnVi/jwlSVKuj92Rrlsoe9Kl\npTQRmZXm3LyLgd0ih8/2mwW/VFcjRyPJPFazDgQC3TyZRsDKErxuy/48k8kgFApZbhWph4iJRNmy\nVpIV2IhYluVu5K+mo5YSEev5nrWcE6zPNp8GbgRjx441fA7snMxCSyfWOk9JkvDLX/4Sn332GRKJ\nBNra2ixv2XPw4EHMnz8fhw8fhs/nw/XXX49FixZZGlMLZU+6FJH5fD4lYjQLu9wDag+TmoOA/p3m\nbIQA1KLRdDptOlJhl3pqum0mk1FWEbFYzDFPphYRU/EFAIVwM5lMXo0Y6ErosdeXXdaS40Itaiz1\nwg7+Puf1U7ZZDH2PTr9o7B5X7TwTiQSCwSD69OmDTZs2YevWrejfvz8aGhrw3HPPYcyYMaaOFQgE\nsGrVKkycOBHxeByNjY2YNWsWRo4cacep5B7L9hFdBvUaiMfjln17bKRnxj2g9rv5HAQHDx7E66+/\njnnz5uWMke+BZ+WOQCBguViC1W19vq5tfOjhFYSu0lEt3dYN0PUTRRHhcDjHV836ZEnnVVtu80TM\nLuF5ImYlFJJr3ExomYWWfkrVlnTf5Os58c477+Dss882vCO2Wx5d4NR5fulLX0IikcAll1yCBQsW\n4IMPPsDgwYNNj9u3b1+lL28sFsOoUaNw6NAhj3TVwCYlzH75/BLdSt9Ydh6FEm8DBw7MIdxC0GoJ\nqXbsQqAHklwYkUhE2cWY7F7UsF2vbmsn2Oi2qqpKkToIWq4APUSsphPzREzWP7X2gnY4C9wgKppb\nMBhUrlO+RFZ7e7uymjFzfm7cH2xA0t7ejmHDhsHv99tKjh9++CF27NiByZMn2zYmi4ogXfqv0RuZ\ndyRQBZUdCYlUKlUw8aYGtfPQ20RdD/hImchMFEX4fD4lQQZA0YfdjvBo6W80ujZCxKy+yxMxKxGx\nWzHZ7Sygcd1EvkTWpEmTTJ2fm1IMe6yTJ0/ankiLx+OYO3cuVq9ebakPSz6UPekSjEZ5ag1paBlr\nFrQcTaVSuhJvFHGwuhV7HkY7ihW6BnykTNooq/eJoqhsKEnzo2iIfof9Y+fDRhY0iq7tSFwaJWIi\nV1bbZa8p2bxYks6XcS/1fhOAdeeEW+Dv7fb2dlsbmIuiiLlz5+Kaa67BnDlzbBuXR9mTLhvp8tod\nD9LptLYhN/tgsJGoEadDKpXCmjVr8K1vfSvn30nHpL6hRpqoq5EuXz3H+m1Z3ZbmrhZZskRFDyBr\n77JCxPSiot0/otGooySlRsRsNSG9cGj7JJY8edcEQY2I85U5FyMyLPR7W7ZsQVNTk/L7RpwTwKke\nxE6fI41rt0/3uuuuw+jRo7F48WLbxlRD2ZMuwefz5c3ck/0rn1XLqEShFom2t7fr/nw4HM4hXHpg\nE4mEsvS34sbgdVs1v61e3ZYlKnJe8PYuo0RML0HqUOakK0ILJDHRlkzsfcHqn+wfAN2iWDUiDgQC\nOTsm80QMuEdUekD2yHxQcxSIoqjIaVolwHY4J/gXSGtrq22FHG+++SaefPJJjBs3Dueeey4EQcDK\nlStx6aWX2jI+i4oh3XxRnl49VC/p8lowG4maTeixNrVwOGxq1wU20uLbQQK5vQ+IaKz4bbXsXXqI\nmOZAL0E7dk8wAj2Ezy671SqqeCJmNWLWU8yOR0RMFXUULDjVb8JI0cL06dNNH4eukZ6eE2rOCTOw\nM9KdOnWqabulUZQ96Wol0ozqoWpj8GCTUFpJMqOkS+RH3c8oojQLWZbR1tYGAAqRsFEY+W2diiz1\nEDEVs9DvsqTlRqRHcosZwlcjYgA5GjH94RNRdG+wjYHoGhBJs2OVc7tILS1dyzmhxxnCv0Ao/1Bu\nKHvSBXI7jaktqfUSSz7CZDdszKfZGomWOzs78eijj+KrX/0qzjjjDEVfNQMiEkmSlPmxui39PJ9u\n6xToAaSETDAYVJbdZqUJM2ClBDsq6lgQaeghYgIrZfBRllp1nVaZsxYRG3n5d3Z2Yu/evRg3bpyh\n86bj6I2mtZwTepwhdN7suZX6y0cNFUG6BEmScPLkSVt7L1CChSq1rPQ0ALo7J2666aaceRqNlNmX\nDJEIRY+sbkvyitt+W+DUNVQjfDPShFEiLpZ2zBIxrZKSyaTig2WX3YU6sAHqRCxJubsBqy3b9Vwr\nGscMrFjGjDonAGDHjh3Yvn07gsEgMpmM6bJ3woIFC/D888+joaEBu3btsjSWHpT1bsCEeDyORCIB\nSZJQW1trWh+UZRknTpxAjx49IMtyToctvdVpWjsCA7nRstayNt/n+bmyum0kElGkBdbulM1mFQ+y\n22SrVU1mBGr6qREiZm1o9NJxG/TS8fl8CIfD3YIBrXNUI2L+eWWX4bSioaiRSJuuj1UNVQtu7QSc\nTqchiiJ27NiBX/7yl3jttdfQ2dmJkSNH4uabb8Y111xjatw33ngDsVgM8+fPt5N0NW/0ioh0KQqN\nx+OWls1085qVJ2iMfMUNfLRMRQlGjsGSN6/bxmIxRbelcelmddJjy4JeCFrVZEZgJllHGmkmk3FE\nStCLfM4IFnrOUYuIeR8xLcupax1F92oaqpl+tlrn6da19fv9uOCCCzBx4kR87Wtfw7PPPotdu3ah\npqbG9JjTpk3DgQMHbJxlflQE6Uaj0YIe3UKgZT/QRZJWtmFnG41QMi8Siagm8/74xz/i/PPPx5ln\nnql8Xmv1wTsx8um2sVgsp/RTjaR4ErYjQWO2mswICpEU2/iHXkiZTMa1JBQrZ1CfZKPHNELEarou\n/ZzuJ77xD6uhZrNZtLa24uTJkzj77LNtKXN2AmrVaNFoFFOmTCnyzIyhIkiXXV4RmegFXzBBS3+z\nZEFz6Ozs1NVI/corr1T9PAu1pjms3xZADhnz2l+hB5h8llaI2IlqMiMggqDtb4jw7TxHPWCvg91W\nOCNETGCj63wd2MiuCEA1mVXIVeC2v7pcG5gDFUK6BNZ2owdqvW3b2tpM+WwBKHYg2pjRju3S+XJl\nQN1va7SSyy4idruaTA3sHHgpwemXjZ45OAn2e6Q5kKxDHmA9Hdh69OihFBqwL21aSeVzFZh9XoyC\nJfeTJ0/aWgLsJiqCdLW8ulrgy2LZJJlR9wCBomW6KfVqTKxHlT1+Pt0WgFK6a2c2Xo2I2Qwyu808\nacKiKCIQCBSl7SOQK2fouQ5ORP1soqwYVXX8HGpqalTLd410YCOCpX+jlwibrHN7J2BWXrA70uW1\ncSdREaRLKESY+Xrb6h2DB0vgFOUlk0ndn3/ttdfQu3dvTJw4EcApuYOKJfLptm5UctEDxB6HiI6I\nKZvNIpFIqCbqnIr2SMKxQ84wS8SCICj6cbHseEaSdWrFCkTEBw8eRCaTQd++fXOiWPos/S47HhEx\nlTLT/enUlkLsc2nnVj3z5s3DunXrcOzYMZx55plYsWIFWlpabBlbDacF6bLL9EIaq17S1SJw0hT1\nYubMmTnjUdRWW1urqduatV9ZBXk51RwBbBTFJrLUokUrcEvOyEfEFOGx50jfu5sVY9RPxOxqhyVi\nemHU1taqFnWwHmA2IqZiDSJotvAFUN/N2UqZM/1ua2srevbsaeh8tfDb3/7WlnH0oiJIV0saYJfp\ntOQvFBkWIt1CBG40UubHi0ajSCaTSKfTyg1ZCpopWcC0tuvhK7LYkk8iYlEUcx50o0RsVEqwG2yy\nDjhVZu12so71P9NqyCrIPQMgb0TMShMEsuipRcTAqWReISIuVObMygvt7e0YMmSI5fMuBiqCdAns\nF1tolwU9Y7DgXQ5aBG6EdDOZjLLNEOlw5Nul3gBAF6GFQqGi1JnnqybLByIonojZh5d2ny1U6EBS\nAmuVcxulkqyzakUzA16aoAQ0fbek7xLZ0vdJ378aEWv1m+A3ImWJmCVdOzuMuY2KIF020s1ms2hv\nb1ceUKNZZDXSNLtRpRZYv+0HH3yA9vZ2zJgxQ3kwQ6GQkrAigqGHF7B/ya4GO6rJeBTSFVnfKJsZ\nF0WxaFE+cCrC9vv9RUvWsRq2E1r+vn37UFNTgz59+mj+DvviUbsn+NUN/QEKd2AD1ImYLXMGuiS2\nNWvW4NixY7bdC3/+85+xZMkSSJKEBQsW4NZbb7VlXC1UBOkCp3azpSWXnq5iamB9spIkGS4FJtJW\nq9LhO59VV1ejsbExJwro7OzMS3S8dsov2QOBgKVqMzuryfRAi4iJ6AAUpaoOsHcZb5aIafXjtLzE\nRqpaPy/k0NBa3RQiYnYsnoj56rpMJoOPP/4YGzZswB/+8Af06dMHM2fOxM9//nNT5y1JEr75zW/i\n1VdfRf/+/dHc3Iw5c+Y4siEloSJIV5IktLa2Kje10d1MWRDpUnGDmU5lPHjdVs1vy+q2+YhOTTtl\niTiZTOpasqvBjWqyQuCJjhI87Hmq7VxBLxs7tFO3XjyFiJiCCOCUB92pyroRI0ao/jtLdmZePHqI\nmNWI1Yow6HsHuqTCe++9F1/5ylfw1ltv4ejRozh06JDZ08bmzZtxzjnnKDsJf/WrX8Wzzz7rkW4h\n+P1+1NXVKXYqs2CLGyiTa4Z4iCTIUsTqwGwkDXQ18UgkEkp0biUDzZ4HHxHnI6hiV5PRnPXu/qu2\nc4Vd2il7DxXjxUPnSURELx72heNGsg44Fd3qlVX0Qo2IAe1WmPQ8bdmyBX369MGuXbvw7rvvIhqN\nYsSIEZovDD04dOgQBg0apPx94MCB2Lx5s6XzK4SKIF2gi3jJ0G0GfHGDlZ1AKcNNiSLS4Hi/bTKZ\nxMGDB/Huu+/iS1/6kunjqR1fjYjVCIpuaFohFCO6tZKss0s71et3dRpay3jeK233C2fHjh2orq7G\n4MGDEQwGLUW3ZsGv4kRRRCKRUM79mWeewcsvv4wjR46gubkZ3/nOd/D973+/7BJqFUO6gLlqMjap\nFYlE4PP5FNeDGRDxU/Sq1ieB1W1HjBjh6FKGwBIULVPp4aYXAvvS4fVhJwjIqWSdUSIGulYcxdqn\njeZI94UeorPzhZNOp7FhwwZ0dHSgZ8+emD17NqLRqGvuCB7sC5AClhdeeAHvvPMOHnvsMTQ2NmL7\n9u3YunWrqW2tWAwYMAAfffSR8veDBw9iwIABVk8hLyqGdFnDth7wSS1KvNGSxihY3RZATsMVgl7d\n1knQ8lmWu5rC8FlwN4oceN+vG8k6tfJmPuJnl9OsDu7k3EjSov4aTrbBpNJx9sXq9/tx+PBhxZ1z\n/PhxdHR0oFevXnadoiGw30FNTQ3a2trw7W9/Gz6fD6+88ooS1V588cW4+OKLLR+vubkZe/fuxYED\nB9CvXz88/fTTeOqppyyPmw8VQ7pAfucAgd6iWh3AzBQ38AUYiURCIRQyz2v1SaDPW+1+Xwj5qslY\nFCpyIMmEJAyKhvU6CcxKCXaCrjnpx+RKcauqjuC0DQzQ108jEomgvr4eJ0+eRI8ePdCzZ08lGetW\nYMAn7AKBANatW4c777wT3/nOd3D55Zc7Mhe/34+f/vSnmDVrlmIZGzVqlO3HYVERO0cAp9rRHT9+\nHD169Oj2BfHkqNW+UZa7do/QU2Ko5t9ls85sSXAgEEAwGOy2vGtvb8fvfvc7LFy40Iar0B18VBkK\nhWwpxVVLeuRzTDghJZgBS/qFNOxCdiezRMwnDfXuSmI32O/E7/fj2LFjqK+vV1ZofETsRLIOyNWx\nI5EIOjs7cfvtt+PYsWN4+OGH0bt3b1uP5xI0L1LFkC5le0+cONEtelVr4agFIl014ibwzcmp1pxN\nTNHNTCTHZp8pceV0gYMRgrEKtVJRckyQbEPJumJppnYkyrTsTnq90qw7oliJS+BU34ZgMKi5lZOW\nm8AuIuaLLQKBADZt2oTbbrsNixcvxrx584ryMrIJlb1dDwtWHqAbnAR5PdVprDasFi2z0oRZv22+\nZazR5boaihFVqjkmiPQlSVKSde3t7a5ETwRWM7UjUablOy3klaYih2JuHwScujdYV40WeKmJPs/6\niNU0Yj3fKSWwyaWRTqdxxx13YM+ePXjmmWccT2YVExUT6VLU0draikgkoiRJQqEQIpGIoRv8xIkT\nOR5dXpqg8bT624bDYUMPNruPld7luhpKZdmaL6pUixLpoeVfOFbnzmqmbm9KyRIxX+RgtGjFrvmw\nfRvs3KjUSEQMoFsp8c6dO7Fs2TK0tLRg4cKFRVkJOYDTI9KlF0gikUAwGDS8qSSBjZbVpIlsNtvN\nb0s/N/NgP/roo1i4cGFOMk2rAksroiiFajL+wVaLKtnEDu0ea3cCi122FqtnA0XDfL9dvuLMbPWg\nETidsDMSEQNd1+a1117DiBEj8Mwzz2DTpk148sknMXToUFvnVaqomEg3lUqhtbUVkiQhFApZ8u+1\ntrYiHA4rZFdIt3VrCc8ndUgfpvlQdFuMSIHVKkmfM4t8uimvhfPXnE/KFCtqYnvd5lv50D1F50gv\ndPJPW5Fg2JdgsVc+6XRaeRlns1nMnz8f27dvRzwex5QpUzB58mSsXLmynDVcHpUf6dJDRn1ozYIe\n+EQigXA4bKlPgt3gCxxoPyx6KCVJQjweB2CfPlwIrJRgl1apRzflo0TSTFmtstQ1U8BY9aAR3ZSN\nbou18qF5ULERVXk+9NBDSCaTWL9+Pc444wxs3boV+/fvryTCzYuKiXSpyxjtK2a06Q2r28qyjHA4\njKqqKlXd1u/3IxwO23YjUzWUnpcFG72oRVFm7FxmUGgeboDIKZ1OK/0yZFm2JUo0MxenNFMan/9O\n1cVMe0MAABfySURBVIiYijxKIbql60Ev43/84x9YtGgRPve5z+HWW28tSo8PF1H5kS7dWGZKgXnd\nlm+ZSCWaVnTbfPjTn/6ECy+8EH379s37e2w1mdY8CjXA0aMPF4KeebgBiirZ6jonmuAUAl/l50RU\naaTaDIBSmMPqqG6Bomy6HoIg4Fe/+hWefvppPPTQQzj33HNdm8uePXtw5ZVXKrywf/9+3HXXXVi0\naJFrc+BRMZEuqxtls1lUV1cX/AzfL5d0W4qc2JJgKm7QG5HaCb3VZEagpQ/n8w87ISWYnbuR7c7p\nXFndVO1cjTomWLdIsa8HWwATCAQKFnM4Ud6spiF/8sknWLRoESZOnIg777xTSZ4WA5IkYeDAgdi0\naVNOZzGH4EW6LFi/LfXLpQcTQI59i25iki+o2KBQQscO8A+T3e311MpDtVwE5JAoZlMYwNx250bP\nVY9jgnymxSxnpnmotaLUchI4Vd7Ma8iCIOCpp57CmjVr8MADD+D8888vuma7du1aDBs2zA3CzYuK\nIV0C759lQSTGNtQg2xeBbmK/36/6MOVL6JjVEROJBI4dO4a+ffsqtrFi9ChQ67tAy1bKqFO7Pbe9\npnbu4gDk7zHBtuXkX7A+n09ZdRSznNlIlF2on4ZVIqZghOx5R44cwdKlSzFw4ED89a9/tdwJzC78\n7ne/w1VXXVXsaVSOvABAkQUSiQTq6upyfsbqtmxjaLJcsTqUUSN9viQH/8CqzfnnP/85UqkUevXq\nha9+9auKhajYD7XaEp7Xh7USOnZppk4nqAodW620GTi1GaWbBQ4EJ0qJ+ZcOnXO+xkasU4Pm8dxz\nz2HVqlW455578LnPfa7o0S0hk8mgf//+eO+999zq5VD58gKBlxdY3ZaKG+jmohuC+pia1eWMLF95\nWeLIkSNIp9OQZRnHjh3Dxx9/jEGDBhW1l2m+slk2Uae1gwMldKz2l9BaOrsFOldyBNAL2e8/tbOD\n1aSkETipIRu16dEKsbW1FXV1dWhvb8fy5csRDoexdu3abkFPsfHSSy+hsbGxJJrnVBTpsn0TCum2\nABzt6VpoLzO6gcPhMHr06KE02RkwYADC4bBt8zACs5VLRjXTQhGi0USZU+CjbPYesdtXWwjsC8gt\nTV3NCUP3iCiKCAQC+MMf/oCVK1ciHA5j4sSJmDNnDv75z3+WHOk+9dRTJSEtABUmL1CN+8mTJxWN\nlZajrM5Lpbs+n89Wv61RUFSYTCaVXqZESIVkCTvhBsnp9Q/TkrVY3l+C1b4Nakt1wLhmynfiKpbc\nBJyS6KhbXDwex3e/+10kEglcf/312L9/P7Zs2YJLL70Ul19+eVHmqIaOjg4MHjwY+/fvR01NjVuH\nrfzWjkCXTNDe3o5sNotYLJZXty3WBoxA994ArIFdjZgA5yrM2J4NbpfN8vpwJpMBAEtJSTvm5FTT\noHzfrRoRk0Oi2CXNrFWQXkBvvvkmvve972Hp0qWKD9ZDDk4PTZeST4lEQtGc6GYoFX8pW8WltkzU\nK0tYJSY2kivWC4g0RGr4Tt8NS8R26cN64LSGrOUioGiYdUxQkFDMfhpA9+1zkskkvv/97+PAgQN4\n9tln0a9fP9fm0traioULF+Lvf/87fD4fHn30UUyePNm149uFiop0qZIskUhAFMUcwZ92TSi2n9KM\nO4KHFbdEvijbbejNwjsd/ZeKhgwgxwtOcosTZdyFwEe3wWAQW7duxfLly3HDDTfg2muvdf1FcO21\n12LGjBloaWlRpA7qjVKCOD3kheuuuw6ffvopzjvvPMRiMbzzzju4++67EY1GNauQnL5xnKgm0zpO\nIWKiB6mUlqtmdEo7+0uUSleyfNdEb98Fu2QY/pqIooh7770X27Ztw89//nOcddZZlo9hFG1tbTj3\n3HOxb98+149tEqcH6cqyjLfeegvf+ta3cPDgQUyfPh2HDh3COeecg+bmZkyZMgXDhg0DcGp7H/ZB\npRJfu/yldu9NZvT47DKdbQPJekzd1EsB/S0PjcKof1hNpyxmgoqW8HqviR4ipnPSe15qSbv33nsP\nN998M6688kp84xvfKNpLaefOnbjhhhswevRo7Ny5E01NTVi9erXhxlYu4vQgXQB4+eWXsXv3btx0\n001K787du3djw4YN2LhxI9577z2EQiGcd955aG5uxqRJk1BfX69647LEZARu7k2WD3xSiNVLjcoS\nVlGMXRy0+ktQG0y/31/074dfwluB0UQdCz5pJ0kSfvKTn2Dt2rV45JFHMGLECEtzs4qtW7diypQp\n2LBhA5qamrBkyRLU1dVhxYoVRZ1XHpw+pFsIsiwjHo9jy5Yt2LBhAzZt2oTDhw/jzDPPRFNTEyZP\nnowxY8bA5/MZ1g9LZcdbIHeJmM8W54ZeWgpbCAG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SdXV1slMhEokgl8sVdBij6IddirLZdFrisi8otajRzM6xTkLNe6pW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\nJ2L06NHw+Xy69cNK2fEWKFwiFrPFOaGXVsIWQsC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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -415,7 +486,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Norm\n", "The norm of a vector $\\textbf{u}$, noted $\\left \\Vert \\textbf{u} \\right \\|$, is a measure of the length (a.k.a. the magnitude) of $\\textbf{u}$. There are multiple possible norms, but the most common one (and the only one we will discuss here) is the Euclidian norm, which is defined as:\n", @@ -429,7 +503,9 @@ "cell_type": "code", "execution_count": 14, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -461,7 +537,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "However, it is much more efficient to use NumPy's `norm` function, available in the `linalg` (**Lin**ear **Alg**ebra) module:" ] @@ -470,7 +549,9 @@ "cell_type": "code", "execution_count": 15, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -491,7 +572,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Let's plot a little diagram to confirm that the length of vector $\\textbf{v}$ is indeed $\\approx5.4$:" ] @@ -500,14 +584,16 @@ "cell_type": "code", "execution_count": 16, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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Wc4HdbFZyFfxwno/Gp59moqkJSktDR3FTjI+PmxzQzmWSd87TKD95kpV33snpV19lYvXq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\nNRfYzZZuLhHJyZu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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -525,14 +611,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Looks about right!" ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Addition\n", "Vectors of same size can be added together. Addition is performed *elementwise*:" @@ -542,7 +634,9 @@ "cell_type": "code", "execution_count": 17, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -574,7 +668,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Let's look at what vector addition looks like graphically:" ] @@ -584,14 +681,16 @@ "execution_count": 18, "metadata": { "collapsed": false, + "deletable": true, + "editable": true, "scrolled": true }, "outputs": [ { "data": { - "image/png": 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PYGzSWIZfM7zOyzuWf4xDJw+Vfh08eZBiXUx+UX6Znx86eYhTBWfmQV9+3uV0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\n7TJ2H9/NykErQ75Iy1azEKK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5dCcrBq8I+SItW81CiLoKXqF2OMq0x997kH0gm2WDlpHYJrFOq/Si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\nBksAAAAASUVORK5CYII=\n", "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -616,7 +715,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Vector addition is **commutative**, meaning that $\\textbf{u} + \\textbf{v} = \\textbf{v} + \\textbf{u}$. You can see it on the previous image: following $\\textbf{u}$ *then* $\\textbf{v}$ leads to the same point as following $\\textbf{v}$ *then* $\\textbf{u}$.\n", "\n", @@ -625,7 +727,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "If you have a shape defined by a number of points (vectors), and you add a vector $\\textbf{v}$ to all of these points, then the whole shape gets shifted by $\\textbf{v}$. This is called a [geometric translation](https://en.wikipedia.org/wiki/Translation_%28geometry%29):" ] @@ -634,14 +739,16 @@ "cell_type": "code", "execution_count": 19, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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EmzaVpK2UA7RGHYecOBi3oKSEpMQ4e6OVny/dFwsWwIwZsurviy+k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\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -678,14 +785,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Finally, substracting a vector is like adding the opposite vector." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Multiplication by a scalar\n", "Vectors can be multiplied by scalars. All elements in the vector are multiplied by that number, for example:" @@ -695,7 +808,9 @@ "cell_type": "code", "execution_count": 20, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -724,7 +839,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Graphically, scalar multiplication results in changing the scale of a figure, hence the name *scalar*. The distance from the origin (the point at coordinates equal to zero) is also multiplied by the scalar. For example, let's scale up by a factor of `k = 2.5`:" ] @@ -733,14 +851,16 @@ "cell_type": "code", "execution_count": 21, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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s8qRqqi8dqaYH169nk9/Psq5d6d+8OXPbXsE5iyew85k34cYbq3xvmlJM7NAB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\n0APIBpa5raWCphZAdvBxQ4zv74VzlR6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r6Z8bNrDZ72d51670b9KE+a2uofeS8ex+7i249dYK35uiFBPatkVFOiNAJZrs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\nAnoAmcByt7WU0dQUyCx9XBfj+3vhWKW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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -773,7 +893,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "As you might guess, dividing a vector by a scalar is equivalent to multiplying by its inverse:\n", "\n", @@ -782,7 +905,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Scalar multiplication is **commutative**: $\\lambda \\times \\textbf{u} = \\textbf{u} \\times \\lambda$.\n", "\n", @@ -793,7 +919,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Zero, unit and normalized vectors\n", "* A **zero-vector ** is a vector full of 0s.\n", @@ -806,14 +935,16 @@ "cell_type": "code", "execution_count": 22, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -834,7 +965,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Dot product\n", "### Definition\n", @@ -856,7 +990,9 @@ "cell_type": "code", "execution_count": 23, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -879,7 +1015,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "But a *much* more efficient implementation is provided by NumPy with the `dot` function:" ] @@ -888,7 +1027,9 @@ "cell_type": "code", "execution_count": 24, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -908,7 +1049,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Equivalently, you can use the `dot` method of `ndarray`s:" ] @@ -917,7 +1061,9 @@ "cell_type": "code", "execution_count": 25, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -937,7 +1083,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "**Caution**: the `*` operator will perform an *elementwise* multiplication, *NOT* a dot product:" ] @@ -946,7 +1095,9 @@ "cell_type": "code", "execution_count": 26, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -979,7 +1130,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### Main properties\n", "* The dot product is **commutative**: $\\textbf{u} \\cdot \\textbf{v} = \\textbf{v} \\cdot \\textbf{u}$.\n", @@ -991,7 +1145,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### Calculating the angle between vectors\n", "One of the many uses of the dot product is to calculate the angle between two non-zero vectors. Looking at the dot product definition, we can deduce the following formula:\n", @@ -1007,7 +1164,9 @@ "cell_type": "code", "execution_count": 27, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1031,14 +1190,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Note: due to small floating point errors, `cos_theta` may be very slightly outside of the $[-1, 1]$ interval, which would make `arccos` fail. This is why we clipped the value within the range, using NumPy's `clip` function." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### Projecting a point onto an axis\n", "The dot product is also very useful to project points onto an axis. The projection of vector $\\textbf{v}$ onto $\\textbf{u}$'s axis is given by this formula:\n", @@ -1054,14 +1219,16 @@ "cell_type": "code", "execution_count": 28, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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F3op0Q96fQVBOf0UlpxfBHMouWUJn4MQFCwL5+Ibik0/gsstcYS3Xrx889hi0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\nFhEJORVqEZGQU6EWEQk5FWoRkZD7f9FA/lHa1t0VAAAAAElFTkSuQmCC\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1091,7 +1258,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "# Matrices\n", "A matrix is a rectangular array of scalars (ie. any number: integer, real or complex) arranged in rows and columns, for example:\n", @@ -1105,7 +1275,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Matrices in python\n", "In python, a matrix can be represented in various ways. The simplest is just a list of python lists:" @@ -1115,7 +1288,9 @@ "cell_type": "code", "execution_count": 29, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1138,7 +1313,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "A much more efficient way is to use the NumPy library which provides optimized implementations of many matrix operations:" ] @@ -1147,7 +1325,9 @@ "cell_type": "code", "execution_count": 30, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1172,7 +1352,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "By convention matrices generally have uppercase names, such as $A$.\n", "\n", @@ -1181,7 +1364,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Size\n", "The size of a matrix is defined by its number of rows and number of columns. It is noted $rows \\times columns$. For example, the matrix $A$ above is an example of a $2 \\times 3$ matrix: 2 rows, 3 columns. Caution: a $3 \\times 2$ matrix would have 3 rows and 2 columns.\n", @@ -1193,7 +1379,9 @@ "cell_type": "code", "execution_count": 31, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1213,7 +1401,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "**Caution**: the `size` attribute represents the number of elements in the `ndarray`, not the matrix's size:" ] @@ -1222,7 +1413,9 @@ "cell_type": "code", "execution_count": 32, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1242,7 +1435,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Element indexing\n", "The number located in the $i^{th}$ row, and $j^{th}$ column of a matrix $X$ is sometimes noted $X_{i,j}$ or $X_{ij}$, but there is no standard notation, so people often prefer to explicitely name the elements, like this: \"*let $X = (x_{i,j})_{1 ≤ i ≤ m, 1 ≤ j ≤ n}$*\". This means that $X$ is equal to:\n", @@ -1262,7 +1458,9 @@ "cell_type": "code", "execution_count": 33, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1282,7 +1480,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "The $i^{th}$ row vector is sometimes noted $M_i$ or $M_{i,*}$, but again there is no standard notation so people often prefer to explicitely define their own names, for example: \"*let **x**$_{i}$ be the $i^{th}$ row vector of matrix $X$*\". We will use the $M_{i,*}$, for the same reason as above. For example, to access $A_{2,*}$ (ie. $A$'s 2nd row vector):" ] @@ -1291,7 +1492,9 @@ "cell_type": "code", "execution_count": 34, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1311,7 +1514,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Similarly, the $j^{th}$ column vector is sometimes noted $M^j$ or $M_{*,j}$, but there is no standard notation. We will use $M_{*,j}$. For example, to access $A_{*,3}$ (ie. $A$'s 3rd column vector):" ] @@ -1320,7 +1526,9 @@ "cell_type": "code", "execution_count": 35, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1340,7 +1548,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Note that the result is actually a one-dimensional NumPy array: there is no such thing as a *vertical* or *horizontal* one-dimensional array. If you need to actually represent a row vector as a one-row matrix (ie. a 2D NumPy array), or a column vector as a one-column matrix, then you need to use a slice instead of an integer when accessing the row or column, for example:" ] @@ -1349,7 +1560,9 @@ "cell_type": "code", "execution_count": 36, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1371,7 +1584,9 @@ "cell_type": "code", "execution_count": 37, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1392,7 +1607,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Square, triangular, diagonal and identity matrices\n", "A **square matrix** is a matrix that has the same number of rows and columns, for example a $3 \\times 3$ matrix:\n", @@ -1406,7 +1624,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "An **upper triangular matrix** is a special kind of square matrix where all the elements *below* the main diagonal (top-left to bottom-right) are zero, for example:\n", "\n", @@ -1419,7 +1640,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Similarly, a **lower triangular matrix** is a square matrix where all elements *above* the main diagonal are zero, for example:\n", "\n", @@ -1432,14 +1656,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "A **triangular matrix** is one that is either lower triangular or upper triangular." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "A matrix that is both upper and lower triangular is called a **diagonal matrix**, for example:\n", "\n", @@ -1456,7 +1686,9 @@ "cell_type": "code", "execution_count": 38, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1478,7 +1710,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "If you pass a matrix to the `diag` function, it will happily extract the diagonal values:" ] @@ -1487,7 +1722,9 @@ "cell_type": "code", "execution_count": 39, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1512,7 +1749,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Finally, the **identity matrix** of size $n$, noted $I_n$, is a diagonal matrix of size $n \\times n$ with $1$'s in the main diagonal, for example $I_3$:\n", "\n", @@ -1529,7 +1769,9 @@ "cell_type": "code", "execution_count": 40, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1551,14 +1793,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "The identity matrix is often noted simply $I$ (instead of $I_n$) when its size is clear given the context. It is called the *identity* matrix because multiplying a matrix with it leaves the matrix unchanged as we will see below." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Adding matrices\n", "If two matrices $Q$ and $R$ have the same size $m \\times n$, they can be added together. Addition is performed *elementwise*: the result is also a $m \\times n$ matrix $S$ where each element is the sum of the elements at the corresponding position: $S_{i,j} = Q_{i,j} + R_{i,j}$\n", @@ -1579,7 +1827,9 @@ "cell_type": "code", "execution_count": 41, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1603,7 +1853,9 @@ "cell_type": "code", "execution_count": 42, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1626,7 +1878,9 @@ "cell_type": "code", "execution_count": 43, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1647,7 +1901,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "**Addition is *commutative***, meaning that $A + B = B + A$:" ] @@ -1656,7 +1913,9 @@ "cell_type": "code", "execution_count": 44, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1677,7 +1936,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "**It is also *associative***, meaning that $A + (B + C) = (A + B) + C$:" ] @@ -1686,7 +1948,9 @@ "cell_type": "code", "execution_count": 45, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1711,7 +1975,9 @@ "cell_type": "code", "execution_count": 46, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1732,7 +1998,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Scalar multiplication\n", "A matrix $M$ can be multiplied by a scalar $\\lambda$. The result is noted $\\lambda M$, and it is a matrix of the same size as $M$ with all elements multiplied by $\\lambda$:\n", @@ -1757,7 +2026,9 @@ "cell_type": "code", "execution_count": 47, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1778,7 +2049,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Scalar multiplication is also defined on the right hand side, and gives the same result: $M \\lambda = \\lambda M$. For example:" ] @@ -1787,7 +2061,9 @@ "cell_type": "code", "execution_count": 48, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1808,7 +2084,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "This makes scalar multiplication **commutative**.\n", "\n", @@ -1819,7 +2098,9 @@ "cell_type": "code", "execution_count": 49, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1842,7 +2123,9 @@ "cell_type": "code", "execution_count": 50, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1863,7 +2146,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Finally, it is **distributive over addition** of matrices, meaning that $\\lambda (Q + R) = \\lambda Q + \\lambda R$:" ] @@ -1872,7 +2158,9 @@ "cell_type": "code", "execution_count": 51, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1895,7 +2183,9 @@ "cell_type": "code", "execution_count": 52, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -1916,7 +2206,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Matrix multiplication\n", "So far, matrix operations have been rather intuitive. But multiplying matrices is a bit more involved.\n", @@ -1961,7 +2254,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Let's multiply two matrices in NumPy, using `ndarray`'s `dot` method:\n", "\n", @@ -1984,7 +2280,9 @@ "cell_type": "code", "execution_count": 53, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2011,7 +2309,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Let's check this result by looking at one element, just to be sure: looking at $E_{2,3}$ for example, we need to multiply elements in $A$'s $2^{nd}$ row by elements in $D$'s $3^{rd}$ column, and sum up these products:" ] @@ -2020,7 +2321,9 @@ "cell_type": "code", "execution_count": 54, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2042,7 +2345,9 @@ "cell_type": "code", "execution_count": 55, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2062,7 +2367,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Looks good! You can check the other elements until you get used to the algorithm.\n", "\n", @@ -2073,7 +2381,9 @@ "cell_type": "code", "execution_count": 56, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2093,7 +2403,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "This illustrates the fact that **matrix multiplication is *NOT* commutative**: in general $QR ≠ RQ$\n", "\n", @@ -2104,7 +2417,9 @@ "cell_type": "code", "execution_count": 57, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2132,7 +2447,9 @@ "cell_type": "code", "execution_count": 58, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2154,7 +2471,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "On the other hand, **matrix multiplication *is* associative**, meaning that $Q(RS) = (QR)S$. Let's create a $4 \\times 5$ matrix $G$ to illustrate this:" ] @@ -2163,7 +2483,9 @@ "cell_type": "code", "execution_count": 59, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2191,7 +2513,9 @@ "cell_type": "code", "execution_count": 60, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2212,7 +2536,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "It is also ***distributive* over addition** of matrices, meaning that $(Q + R)S = QS + RS$. For example:" ] @@ -2221,7 +2548,9 @@ "cell_type": "code", "execution_count": 61, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2244,7 +2573,9 @@ "cell_type": "code", "execution_count": 62, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2265,7 +2596,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "The product of a matrix $M$ by the identity matrix (of matching size) results in the same matrix $M$. More formally, if $M$ is an $m \\times n$ matrix, then:\n", "\n", @@ -2282,7 +2616,9 @@ "cell_type": "code", "execution_count": 63, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2305,7 +2641,9 @@ "cell_type": "code", "execution_count": 64, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2326,7 +2664,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "**Caution**: NumPy's `*` operator performs elementwise multiplication, *NOT* a matrix multiplication:" ] @@ -2336,6 +2677,8 @@ "execution_count": 65, "metadata": { "collapsed": false, + "deletable": true, + "editable": true, "scrolled": true }, "outputs": [ @@ -2357,7 +2700,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "**The @ infix operator**\n", "\n", @@ -2368,15 +2714,17 @@ "cell_type": "code", "execution_count": 66, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "Python version: 2.7.11\n", - "Numpy version: 1.10.4\n" + "Python version: 3.5.3\n", + "Numpy version: 1.12.1\n" ] } ], @@ -2393,14 +2741,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Note: `Q @ R` is actually equivalent to `Q.__matmul__(R)` which is implemented by NumPy as `np.matmul(Q, R)`, not as `Q.dot(R)`. The main difference is that `matmul` does not support scalar multiplication, while `dot` does, so you can write `Q.dot(3)`, which is equivalent to `Q * 3`, but you cannot write `Q @ 3` ([more details](http://stackoverflow.com/a/34142617/38626))." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Matrix transpose\n", "The transpose of a matrix $M$ is a matrix noted $M^T$ such that the $i^{th}$ row in $M^T$ is equal to the $i^{th}$ column in $M$:\n", @@ -2429,7 +2783,9 @@ "cell_type": "code", "execution_count": 67, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2452,7 +2808,9 @@ "cell_type": "code", "execution_count": 68, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2474,7 +2832,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "As you might expect, transposing a matrix twice returns the original matrix:" ] @@ -2483,7 +2844,9 @@ "cell_type": "code", "execution_count": 69, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2504,7 +2867,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Transposition is distributive over addition of matrices, meaning that $(Q + R)^T = Q^T + R^T$. For example:" ] @@ -2513,7 +2879,9 @@ "cell_type": "code", "execution_count": 70, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2537,7 +2905,9 @@ "cell_type": "code", "execution_count": 71, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2559,7 +2929,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Moreover, $(Q \\cdot R)^T = R^T \\cdot Q^T$. Note that the order is reversed. For example:" ] @@ -2568,7 +2941,9 @@ "cell_type": "code", "execution_count": 72, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2593,7 +2968,9 @@ "cell_type": "code", "execution_count": 73, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2616,7 +2993,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "A **symmetric matrix** $M$ is defined as a matrix that is equal to its transpose: $M^T = M$. This definition implies that it must be a square matrix whose elements are symmetric relative to the main diagonal, for example:\n", "\n", @@ -2634,7 +3014,9 @@ "cell_type": "code", "execution_count": 74, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2657,7 +3039,9 @@ { "cell_type": "markdown", "metadata": { - "collapsed": true + "collapsed": true, + "deletable": true, + "editable": true }, "source": [ "## Converting 1D arrays to 2D arrays in NumPy\n", @@ -2668,7 +3052,9 @@ "cell_type": "code", "execution_count": 75, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2690,7 +3076,9 @@ "cell_type": "code", "execution_count": 76, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2710,7 +3098,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "We want to convert $\\textbf{u}$ into a row vector before transposing it. There are a few ways to do this:" ] @@ -2719,7 +3110,9 @@ "cell_type": "code", "execution_count": 77, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2740,7 +3133,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Notice the extra square brackets: this is a 2D array with just one row (ie. a 1x2 matrix). In other words it really is a **row vector**." ] @@ -2749,7 +3145,9 @@ "cell_type": "code", "execution_count": 78, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2769,7 +3167,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "This quite explicit: we are asking for a new vertical axis, keeping the existing data as the horizontal axis." ] @@ -2778,7 +3179,9 @@ "cell_type": "code", "execution_count": 79, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2798,7 +3201,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "This is equivalent, but a little less explicit." ] @@ -2807,7 +3213,9 @@ "cell_type": "code", "execution_count": 80, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2827,7 +3235,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "This is the shortest version, but you probably want to avoid it because it is unclear. The reason it works is that `np.newaxis` is actually equal to `None`, so this is equivalent to the previous version.\n", "\n", @@ -2838,7 +3249,9 @@ "cell_type": "code", "execution_count": 81, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2859,7 +3272,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Great! We now have a nice **column vector**.\n", "\n", @@ -2870,7 +3286,9 @@ "cell_type": "code", "execution_count": 82, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -2891,7 +3309,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Plotting a matrix\n", "We have already seen that vectors can been represented as points or arrows in N-dimensional space. Is there a good graphical representation of matrices? Well you can simply see a matrix as a list of vectors, so plotting a matrix results in many points or arrows. For example, let's create a $2 \\times 4$ matrix `P` and plot it as points:" @@ -2901,14 +3322,16 @@ "cell_type": "code", "execution_count": 83, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2928,7 +3351,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Of course we could also have stored the same 4 vectors as row vectors instead of column vectors, resulting in a $4 \\times 2$ matrix (the transpose of $P$, in fact). It is really an arbitrary choice.\n", "\n", @@ -2939,14 +3365,16 @@ "cell_type": "code", "execution_count": 84, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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FgAHw05/C/PlhPZmBA2u3BaMeukgFmTYtzLyU3A44IDxyWbEirCuz777ljamc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\nAABJRU5ErkJggg==\n", 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ZuBDuvz8sQ7BsWbiQeuONYXmCaqGLogmnXmmWcpHVVS5mzAj3Da2FYp7vcVFf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\n4TjnLfdIPwAAAABJRU5ErkJggg==\n", "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2963,7 +3391,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Or you can represent it as a polygon: matplotlib's `Polygon` class expects an $n \\times 2$ NumPy array, not a $2 \\times n$ array, so we just need to give it $P^T$:" ] @@ -2972,14 +3403,16 @@ "cell_type": "code", "execution_count": 85, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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83EIOfRrwmqouDGx7gcGqesxdhyyHbkz0XHTR5bz//oP4bxsQqj3AQrKzy4Bd\n3HzzTdx+eyEDBw4kJcUmwblFpKctCi3fpX8pcGvgpJcBB5vrzI0x0bNjxw4++mg7EG6pfy5wH5WV\nG6msfJ2//vUUhg0bx8kn9+Lf//3nlJeXYxdh8SOUaYvzgfXAuSKyS0RuF5HxIjIOQFWXAztF5CNg\nOjAhqhEniKbphmRmbRHU1rYoKZlLQ8MYoD2l/ufQ0PAIVVXvc+DAyzz9dAqDBl1Pjx4X8KtfPc72\n7dvbcezw2fsifK3m0FX1hyHsMzEy4RhjwqWqzJw5h8OH50foiAJcTH39xdTX/4bq6reYNKmMP/zh\nCnJzc7nrrkIKC8fQo0ePCJ3PRIqV/hsT5958802uvvp2qqq2Ed31S+sBD5mZC1B9kby83owbV8hN\nN43m29/+dhTPa+xeLsYkiTvumMBzz+XS0PCLGJ7VB7xKx45lHDmygn79BjJuXCHXX38dJ554Ygzj\nSA52LxeXs/xgkLVFULht4fP5WLRoEQ0Nt0QnoBZlANdQU1OGz7eH9etvZeLEFzj55B4MHXojL7zw\nArW1te06g70vwmcdujFxbPny5aSk9AZ6OhhFJ6CQqqql+Hw7WblyOHfcMZWTTurO6NG3smLFCurq\n6hyML3lYysWYOHb11TewatUIoNjpUJrxGfA82dllqH7E6NGjuf32QgYNGmRz3MNkOXRjElxFRQXd\nu/fC5/uUyC4uHQ07EVlAp05lpKdXcMstY7jttkL69esX97f5jQXLobuc5QeDrC2CwmmLBQsWkZo6\nFPd35gC9UH2Iqqr3+PLLV5k6tQMFBWPIzT2Phx76Fdu2bTvmJ+x9ET7r0I2JU1OnzqGm5sdOh9EG\nvTly5Amqqz/is8/m8cc/VtGv31WcdVY+v/nNJD799FOnA4xblnIxJg7t2LGDCy8cyKFDe2hfdahb\nHAH+TmZmGfA3zjrrXMaNK2TMmJvp1s3uxm0pF2MSWGRK/d0kFSjg0KHpHDq0l61bf8FDD23gjDPO\n49JLf8CsWbM5ePCg00G6nnXoDrH8YJC1RVAobREs9Y/HdEsoTgBGUlNTjM+3l7ffHse9977CKaec\nwZAh17Fw4UJqamqcDtKVrEM3Js689dZbVFenAf2dDiUGOgI3UV29GJ9vF2vWXEdx8WxOOqk71113\nCy+//DKHDx92OkjXsBy6MXHGmVJ/t/knR+e4Hzmyjeuvv4E77ihk8ODBpKamOh1cxNk8dGMSkM/n\n46STcqnF/f2MAAAJeUlEQVSufgdnq0PdZBciC8nKKiMlZR+FhTdTVFTIgAEDEmaOuw2KupzljYOs\nLYJaawt3lPrHiifE/U5H9QEqKzfx1VdrmDEjhyFDbuWUU87mP/7jF2zZsiWaQbqKdejGxJGpU+dQ\nWZmog6GRkEdDw2NUV3v55z+fZ8qUw1x22QjOOONCHn/8v/j444+dDjCqLOViTJyIr1J/N2kA1pOR\nUYbI8/Ts2Yu77ipk7Nib6d69u9PBhcRSLsYkmPgq9XeTFGAQPt8zHDq0F6/3cR55pJwzz+zNJZd8\nnxkzZlJRUeF0kBERUocuIsNExCsiH4rIz5t5fbCIHBSRTYGvRyIfamKxvHGQtUXQ8doifkv928oT\nhWOmAUOprS3B5/uMd9+dyH33raR7914MHvxvzJs3j6qqqiicNzZCWSQ6BXgaGAr0BgpFJK+ZXdep\nat/A1xMRjtOYpLZjxw4++mg7MMzpUBJIJnAD1dXP4/PtZt26sdx993y6ds1l5MgxvPTSS/h8PqeD\nDEurOXQRuQx4VFWHB7YfBFRVJzXaZzDwM1Ud1cqxLIduTBv88pe/5ne/+4LDh//sdChJ4Avgb4E5\n7u8xatR13HlnIVdeeSVpaWmORBTJHHou8I9G27sDzzU1UETKRWSZiFwQYpzGmFYkfqm/23QFxlNZ\n6aGmZgsLF17IjTc+TJcuuRQXT+SNN96goaHB6SCbFalB0XeB01U1H3965qUIHTdhWd44yNoiqLm2\nSK5S/8Y8TgeA/9r1PiorN1JZuYrZs/czaNAgunXrxdq1bzgd3DFC+f9hD3B6o+3TAs99TVWrGj1e\nISJ/EZEuqnrM0HFRURE9e/YEICcnh/z8fAoKCoDgm9m2k2v7KLfE4+R2eXn5Ma+Xli4KDIauxa8g\n8N2T4NvlDpxfgQsAL7CE1NRdZGVVU1/vpbZ2DyedlEt+/o307ZvH3r278HjqovJ+8Hg8lJSUAHzd\nX4YilBx6KvD/gSH4Fwl8GyhU1W2N9ummqvsDjwcAi1T1mCgsh25MeKzUP1rqgI/xd9xeOnb0kp7u\n5dAhL2lpKfTqdT4XXZRHnz55nH9+Hnl5efTs2dP1OfRWo1PVIyIyEViJP0UzS1W3ich4/8s6Axgt\nIj/B30q1wJj2hW+MgWQr9Y+Gg/ivR72kpnrp1MmLqpeamp107Xoa55zj77Qvvvhy8vLuJC8vj65d\nuzoddJtZpahDPB7P1/9qJTtri6CmbXH11TewatUIoNixmJzjIZgOOZ4G/PM2/FfbmZleMjK8HD7s\n5ciRSk4/PY/evfPo2zePCy7wX22fffbZZGZmRjH2yIrYFboxxhkVFRWsW7camO10KC5RC3wIeBHx\nX22npHiprf2QrKxvcdZZeXznO3nk5/cmL+9G8vLyyM3NTZg7LobCrtCNcam//GUaDzywhpqaRU6H\nEkOK/17n/qvt9HR/fru+3ovPt49TTz2LvDz/1faFF/qvts877zyys7Mdjju67H7oxsS5iy66nPff\nfxA4br1enIqvQUmnWYfucpY3DrK2CDraFjt27ODCCwdy6NAe4nsh6FAHJf2dduNBSXtfBFkO3Zg4\nVlIyl4aGMcRHZx7OoOQtcTkoGS/sCt0Yl1FVTj31HPbvnw8McDqcRkIdlAxebSfboGS02BW6MXHK\n2VL/cAYlR5KXd39SDErGC+vQHWL5wSBriyCPx9Oo1D+aV7bhDEoOcWRQ0t4X4bMO3RgXOXz4MIsW\nLaKh4Z0IHTG5KiWTneXQjXGRF198kdtum0Jl5drWd/5ay4OSDQ1V9OhxXtxXSiY7y6EbE4emTp1D\nZWVL9z23SklzfHaF7hDLDwZZW/hVVFRwyik9qKvbBOzj6KBkhw5ejhxJvkpJe18E2RW6MXFm69at\n1NXV0LHjd78elOzb17lBSRN/7ArdGBf58ssv+da3vuV0GMZlrPTfGGMSRCQXiTZR0HT5tWRmbRFk\nbRFkbRE+69CNMSZBWMrFGGNczlIuxhiTZELq0EVkmIh4ReRDEfl5C/s8JSLbRaRcRPIjG2bisfxg\nkLVFkLVFkLVF+Frt0EUkBXgaGAr0BgpFJK/JPsOBs1T1HGA8MC0KsSaU8vJyp0NwDWuLIGuLIGuL\n8IVyhT4A2K6qn6pqHbAAuLbJPtcCpQCqugHoLCLdIhppgjl48KDTIbiGtUWQtUWQtUX4QunQc/Hf\n+eeo3YHnjrfPnmb2McYYE0U2KOqQTz75xOkQXMPaIsjaIsjaInytTlsUkcuAx1R1WGD7QUBVdVKj\nfaYBr6nqwsC2FxisqvubHMvmLBpjTBtE6uZcG4GzReQM4DNgLFDYZJ+lwD3AwsAfgINNO/NQAzLG\nGNM2rXboqnpERCYCK/GnaGap6jYRGe9/WWeo6nIRGSEiHwHVwO3RDdsYY0xTMa0UNcYYEz0xGxQN\npTgpGYjILBHZLyLvOR2L00TkNBFZIyJbRWSLiPzU6ZicIiIZIrJBRDYH2uM3TsfkJBFJEZFNIrLU\n6VicJiKfiMj/C7w33j7uvrG4Qg8UJ30IDAH24s/Lj1VVb9RP7jIiMgioAkpV9WKn43GSiJwCnKKq\n5SKSBbwLXJuM7wsAEemoqjUikgq8Adyvqm84HZcTROTfgX7Aiap6jdPxOElEPgb6qeqXre0bqyv0\nUIqTkoKqvg60+otJBqq6T1XLA4+rgG0kcf2CqtYEHmbg/2wm5ftERE4DRgDPOh2LSwgh9tWx6tBD\nKU4ySUxEegL5wAZnI3FOIM2wGf+Coh5V/cDpmBwyGXgAsAE+PwVWichGEbnreDtaYZFxXCDd8gJw\nb+BKPSmpaoOq9gFOA74nIoOdjinWRGQksD/wn5sEvpLd5araF/9/LfcE0rbNilWHvgc4vdH2aYHn\nTJITkTT8nfkcVV3idDxuoKr/ApYBlzgdiwMuB64J5I3LgCtFpNThmBylqp8Fvn8OvIg/hd2sWHXo\nXxcnicgJ+IuTknn02q48gmYDH6jqk04H4iQR6SoinQOPOwA/AJLudoOq+rCqnq6qZ+LvJ9ao6q1O\nx+UUEekY+A8WEekEXA2839L+MenQVfUIcLQ4aSuwQFW3xeLcbiMi84H1wLkisktEkrYIS0QuB24B\nvh+YkrVJRIY5HZdDTgVeC+TQ3wKWqupqh2MyzusGvN7offGyqq5saWcrLDLGmARhg6LGGJMgrEM3\nxpgEYR26McYkCOvQjTEmQViHbowxCcI6dGOMSRDWoRtjTIKwDt0YYxLE/wKYEpYPogYvRwAAAABJ\nRU5ErkJggg==\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2996,7 +3429,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Geometric applications of matrix operations\n", "We saw earlier that vector addition results in a geometric translation, vector multiplication by a scalar results in rescaling (zooming in or out, centered on the origin), and vector dot product results in projecting a vector onto another vector, rescaling and measuring the resulting coordinate.\n", @@ -3006,7 +3442,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### Addition = multiple geometric translations\n", "First, adding two matrices together is equivalent to adding all their vectors together. For example, let's create a $2 \\times 4$ matrix $H$ and add it to $P$, and look at the result:" @@ -3016,14 +3455,16 @@ "cell_type": "code", "execution_count": 86, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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TU5+ytymOT5b+DZ65a8jpdZPsdgXkGsEmGmu5CJcDd6pp4NhqbWPOnKmn+qta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\n805Nblm+XPM9PT2adHzttbB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W09ycyaFDZvbsOcHZs5V0dHSMH+Pe2xvWeuIB09ClhI4O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\nxhiTICyhG2NMgvg/HtxDK78yajgAAAAASUVORK5CYII=\n", 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J8OMfe7Zr8mR4/PGhY+Mkkmu5SCnp6OigodaMpbyRmus3kpKSicHg3a9Fp9NJW9tx1q4t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\nsWPr+COHAog/NPRcoKLfdlX3a4oRULp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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3056,7 +3497,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "If we add a matrix full of identical vectors, we get a simple geometric translation:" ] @@ -3065,14 +3509,16 @@ "cell_type": "code", "execution_count": 87, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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8lHPNaMllNDLV1iORCJ2dPRw54iUadVBeXoPDkfkElIhGMHe3Yz22j7KuNgCS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\n/vjHJ0+XR77zzjtcfPHFADz44IO6Hq2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FEOJ0jfv8+Sn6+voGNuk4QTI5uEmHtbMFKVO5S+YARpOWvAEScWxH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\nAAAAAElFTkSuQmCC\n", "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3098,7 +3544,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Although matrices can only be added together if they have the same size, NumPy allows adding a row vector or a column vector to a matrix: this is called *broadcasting* and is explained in further details in the [NumPy tutorial](tools_numpy.ipynb). We could have obtained the same result as above with:" ] @@ -3107,7 +3556,9 @@ "cell_type": "code", "execution_count": 88, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3128,7 +3579,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### Scalar multiplication\n", "Multiplying a matrix by a scalar results in all its vectors being multiplied by that scalar, so unsurprisingly, the geometric result is a rescaling of the entire figure. For example, let's rescale our polygon by a factor of 60% (zooming out, centered on the origin):" @@ -3138,14 +3592,16 @@ "cell_type": "code", "execution_count": 89, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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ceV4ePPZYbDlzUKn+U6aMP/RG/0YCR4/QNTFPtGdy+ovTCU8/DW+9pQpqXXZZpC0KHCWp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\ntbXhGaiLrzmGuNDQXV29zHroZo4sPJv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6dL2xhPbsqfSVlJE8u4zchZMwF5ow9DqVIxZC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\n", 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FFC4n3vRMPLkFI9zD2MVub2HChGxdez+O0Rr6CJESnn0W7rlH6eYuF/T0DP2e\njAw1j3fNNZGxMVC8Xi+tra00NNg4dMhBT08WJpOKcR+YwORvktObMfJswo7aA2Q8ch8ADV++hfEL\ny0fzUTA21tMx92ScM04Y1X7GIlZrLaeeWkROTk60TdEMIFANXTv0ELBxIzzxhAq+ePhhqKvzv112\nttLbTzklsvYFg8fjobW1lUOHbBw+3InXaybdlIPZ2Un67k2kHNoX9CTnYLT+6QXM2z9QpW//5x6M\nptHfMCZb62j5zOVjLjt040Y1wPjwQ1U3qKoKjEZVeuLIEUhPh+uugzlz/L+/u9uFy7WDJUsq4iKX\nYayhHXqUuP56FYteV6ecd3+MRhXamJUVH7We3XY77TU1tFV/SNeRVjyp+RjziklNG2XCibML7vgR\nAI2nnM+ek8ZxytR5o7bX0GHHm5JK25lfHPW+osVo66FfdBHMnKnuHPvzu9/B88/Dc8/BBD85XE1N\nDcyY4aG8PHYSseLhNxIpQl1tURMAGzfCSy+pH9Prr6tQxltugb47WLNZOfOYp6kJ3nuP5GeeIW/L\nFiZPn8j0pfMprcjHmNJMW9te2tub6O52Bb1rT82Wo86cO35MwaVnhczspI42nFMiFEYUgzQ0QH09\nnHji8evmz1ey4Lvv+n+v19tMQUH0k4k0o0NPioYIrxeuvRZ+8QvIz1evTZ8O998Pv/ylen1Nvz5P\nMTfy8HjgwAH4+GN1j56SotJaeyfIkgGLJReLJReXy4Xd7sBqPUxbGxgMmZhMWUOGQUopaf+f35PT\n/Ane8SUYbvseCDWeCMXoPFFS/UczOv/vf1W4/kknHb9u7161zl8AS6ym+sfcbyQO0A49RJx0EtTU\nwOrVx6/LyIC77orRktx2O+zcqW4vXC4l9JcO7RRTU1NJTU0lPz8Pp9NJe7sDq/UQnZ1JGAxZpKVl\nYuwXA95+oJXsB35GDtB+4RVkV/kZQo6SeE31DyXr1qk7wKl+Ijb/+U9VFaFfMu1RHI5mKir06DwR\n0A49BNTWKmf+9NODR+0N7LEbVX3Q64XDh5UT37dPGZ2Xd7yRAWAymTCZTBQU5NHldNLWaqexsY6O\njmQMhkwaXt7I9E1/A8Bz5y/Jzsk8bh9rdm8e9Sg9UVL9R6Ohr18PJwwI7nE44Le/VYOJP/5RNVfp\nTyyn+msNPXi0Qw8BP/2piir46lejbckwOJ2wZ4+SVdra1K1DcXCZnIMhhCA9LY30tDSKiiQddjvW\nZTdg8PSqfUBXAAAgAElEQVSwJWshk26/irS0tBB8CD/0eJDGZNz548Oz/zhg/35obFQO/KGHVFht\nZ6e6dldVwZ13+n+fw9HGuHEmUkZwMdfEHjrKZZS8845y5LW1yj/GJE1NsH07bFPlarFYVBxbmHBs\n2UvmHTcD0HDT3WQtnITNZqe5uQuvN43k5CxMaekYRGg0KKPNSlf5TDorF4dkf/HISy/Bb34Df/5z\ncOUlrNZPWLAgh/y+iR9NTKJruUQAt1vF9t5/fww682EmOcOFfPRRMleuBMD93EuMT1cjv8zMTEpK\neujo6KC5uZ3WVitebwapqZmkpqaPKvZZuLvjNtU/VKxdq76Ds2cH/h5fqn952OzSRJZYnKaLGzo6\nYNkyuOSS4N8btjoVdrsSU598UsVOOp1qkjPMzrzH0QUXXohYuRJ56aWwciXJ6cfexiclJZGdnc2k\nScVUVExk6lQTaWktrNr0Jm1tVpzOLoK9g1Op/hkJk+o/0louGzao0MRg1LNYT/XXtVyCZ9gRuhDi\nMeB84IiUssLP+iXA34E9vS/9VUr5y5BaGaN4PKrbWdQJ4STnSGh8fQMFD/1MLfzhDwh/mSsDSEpK\nIicnh5ycHPa2NzK5PIXGpiba2z1AJmlpWaSkmIbfT7uNjrknx2gIUWSorVX9axcsCO59breN4uKi\n8BiliQrDauhCiNMAB/DkEA79e1LKC4c9WAJp6N//Ppx+Olw47KcOI/4mOc2D9+QMNdIrOXz9zxnf\nsIGOnPGkr/gjIml0x+7u7sZut9PY2EFHh0SITEymzEHruI/VVH+ATz6BRx9VDr2uTmnns2fDbbcN\n/16d6h9fhDT1XwgxEfjHEA79NinlBQHsJyEc+saNcNZZap4xKnNJEZ7k9EfrXhvmm5cB0Py1m8n7\n8pkhP4bL5aK93U5jo4OuLgNCZJKennU0xj0RUv2jRSym+msGJ9Kp/wuFEB8LIf4phAhiWib+kPL4\njNCRELQ+6PGo0fjLL8MLL6hh2bhxqjBHhJ1587NvHHXmrb97ctTOvHrzZr+vp6amUlCQz+zZ5cye\nXUBxcQ9udx1tbXXY7a0Iuy3hUv0jVQ89HlL9tYYePKGIclkPTJRSdgohzgVeAaYPtvGyZcsoLy8H\nwGw2U1lZeTR5oO8fGMvLt98OGzdWsXp1hI7f0UFVURFs3KgcX3o6Vb0F1fscYdW8eRFZfnvjRlx3\n/y+fdbZRX3Yytd+8ENF+kCrMo9p/H0Ntn5aWxprdu5FScsqsqbTa2vjPpo84nGniJHMemZlmamre\nB3zp833OMZ6Wa2trwn68OXMWYDYL1q5dC8TW76v/ck1NTUzZE8nl6upqVqxYAXDUXwbCqCUXP9vu\nBU6SUtr8rItryWX7dqVRPvUUfO1rYTxQlCc5/XLgANx4IwCN37mLgrMro2cLQEsL3oIC2hctoqHB\nRl1dOx5PJqmpqknHwDruGh+NjQeoqEimuHjsJmLFG6GOQxe9D38HGielPNL7/GTUReI4Z54IfOMb\nat4xbM48jJmco0E+8STi5ZcA8D7/AgVpw0efhB2HA0NVFWazGbPZzPTpPbS1tVFfb6O+/gA9Pdmk\npakmHXrSz0csp/prRk8gYYvPAFVAnhDiAHAnkAJIKeX/AZcIIa4H3EAXcGn4zI0e77yj0quPHAnN\n/o6pU+FvknOYAlmRwO1wkfyVLyEAefHFiKuuCkviQvXmzUdlloDweCA5Gcb7RphJSUlYLBYsFgsz\nZ3poaWnh0CErhw/vQ0oz6ekWMjKyYt65j7Ye+nDEU6q/ruUSPMM6dCnlV4ZZ/xDwUMgsikHCkhHa\n0+MbjUcwkzNQ6l/fTPFDPwag49cPkjG7PLoG9cdmU9pXcrLf1UajkYKCAgoKCnC73dhsNg4ePERj\nYzeQS2amhfT044uEjQW6upqZOzcv2mZowoSu5RIAGzeqmuYvvKBqSo+K9nbYtevYcrXZI+/JGWqk\nV1J79a+ZafuQjlQL6c89hoiRi8xR6urgC19QF8AgcLlcNDfbOHDARkuLF4PBQmZmLqYxUnLX4/HQ\n3r6FpUvnxWx2qMY/ugVdCGlqGmW8eSxOcvqhu7GNlGu+DsCWM65n7m3nRtkiPzid0NWlJjJGMbfQ\n1dVFU5ON/ftt2O0GDAYL2dmWQROYEoGWlkZKSuzMmjU52qZogkS3oAsRf/qTKo8yIpxOpYs/+yy8\n8gpYrWqSc/x4qmtrQ2rnaLH99e2jztx23+MRdeaDxaH7xWaDefNGPVGclpZGaWkJp502jzPOKGfG\nDA/d3bVYrdux2Y7gdnePav8jJZxx6CrVP37kFh2HHjy62uIQbNwIP/6xursPihid5PSL14v32zdi\nOVRHU0kFeQ/9AoshhicOvV6YNCmku8zIyCAjI4OJEydgt9uxWls4eHA7LS0mkpMtZGXlYjTG90+l\nu9tFWpqT7BiS9zShR0sugyAlLFqkqiled10Ab/BXrjY/P2YmOf1SX3/0w3nu+CnGhfOjbNAw2O2q\n5c4Xw5/qL6Wkra2NI0daOHiwDbc7g5QUFeMej/qzTvWPb3Q99FHy3e8qqfab3xxmQ3+TnLE6Gu9H\n/f3PUfz2M2rh+ecxhqubUChpa/PfFDMMCCGOxrhPm+altbWVhgYbhw4dpKcnC5NJxbjHSwKTSvUP\n7Z2NJvbQDt0P27fD8uXwzDODDLBDMMkZdOx1iHB3ukm+7IsUA42nnEfBjwO5/QgvAZ0Lr1eFGEXh\nYmkwGI7GuM+Y4aG1tZVDhxo5fHg/Xq+Z9PRcMjKyQxLjHo449K6uDsxmQUbMdWEZGh2HHjzaofvh\n5JPV38svH7AiRjM5A2Xvv7Yx6eEfAuD4xf0UfGpKlC0KgrY2KC+PeCGygRiNRvLz88nPz2f2bDct\nLS0cPNhAY+M+IJeMjNiLcXc4mqmoiO1CXJrQoDX0AfjtERoD5WpHy8Hv3UfprmocZGB68SmMqXF2\nLT94EM4/H8rKom2JX1wuFzZbCwcO2LDZPL0x7paox7hLKWlq2sTSpbPiIjtU4x+toY+AYzJCUz2w\nJ/I9OUONu9VB8hVfoRTYcdo1zLz9omibFDx+Uv1jjdTUVMaPL2L8+CK6urpobm5h//49WK2QlKSc\ne2pq5GvgxFOqv2b0xIdWECGeeQamFLRzyaTw9+QMKvZ6pKxeTfIVqnKD/f5HY9aZD3suhkn1jzXS\n0tKYMKGYxYvncsYZk5gxw4vHsxOrdRvNzYeHjHEPdRx6V1czpaXxE3veHx2HHjx6hN5HbS1f+GQl\nXzovBbEuFQoKYi6TM2CkRN76PcQnu+mZNgPDvb8hK5Zjy4ejuxumTo22FSOiL8a9rKwEh8OB1Wrj\nwIHttLSkYjRayM7OPdqBKdR4PB6MRju5ueVh2b8m9tAaOsAbb+C98PNQVoZh+f2qrKKUKgQxKysE\nBVwiR/uuI2R/rzfW8oc/hFNPja5Bo8XpVI+vfjVuJp+HQ0pJe3s7hw/bqKtro7s7g5SUXLKyckMa\n465T/RMHraEHyvLleH/wIwyuLtwnnYzh3HNVAPqhQ2oS9NAh5dBzciAztqIXBlL/0N8ofv1xANxP\nPkOyObbtDQibTYUdJYgzB/XjzMnJIScnh2nTvLS1tfXGuNfh8YQuxl2l+heFyGpNPJA4v5Jgcbvh\n6quRd9yBwdWFx2gieVFvpmRamrrFv/BC+PrXoapKyS91dSq7srNz1IcPpYbucbpxXngJxa8/TlPl\nmbByZVw58yHPRRhS/WMJg8FAbm4us2dPYenSeXR3byEvr4mWlk1YrXtxONoYyV1tIqT6aw09eMbm\nCN1mg899DjZuRHR1AZCUnqqKPg0kMxNmzlSPtjYlx2zbppx7UhLk5oIpeh187Bt2kfWz72EE9t/4\nGyaeMzNqtoQcux0KC8FsjrYlESEpKYmcnBwqKqYxe7anN8b9MFaratLRF+MeSAJTe7uNGTMsMd/Q\nQxNaxp6GXlur0sebmtRkWx+pqSrWuaBg+H1ICS0tKkt02zbleJKTlXOP4ERq010Pkb/udTwkIZ9/\ngeS0+IgCYd06eO89FT20fz+ceCJ8+tPHb1dXp/5XM2Ycv87rhT/8QYWVlpQoWWzZsuNH8x98AA88\nAP/+t/p/X3yx+l+53er/nZkJv/gFLFgQlo8aCrq7u7HZWjh40EZTkxshVJOOtLTBMz+t1i2cccak\nuMsO1fhHa+j+eOMNVdipo0M55f6kpATmzEE5D4tFPU44QV0c9uxRzr2rS10ccnPDF2bX0QGXX04+\nUH/2FRR/55LwHCccbN+uHOwjj6jMLacTbrhB3eUsWuTbbrhU/29+U43cH3tMLVdUwI4d8Pzzx253\n6qnqMWWKunA8++yx63/4QyWpbd4Mk2Nz8jAlJYWionEUFY3D6XT2NunYh9UqMRhyexOYfLV44jXV\nXzN6xo6Gvnw5fP7z4HAc78wBpk0b2X6FUBeCU06BK69UtXZnzlSyzsGDytn39Bz3thFr6GvWHK1J\nYP/tI/HlzAGeew4WLvSl4ZpMHJg163hHPFSq/8svw5tvwv/+r++1c8+FiwaJs9+/H/buhSVLjl/3\n6U+ri/DKlSP6OKFmON3YZDJRUlLMokVzWLJkMrNmSaTcjdW6jaamBrq7XTgczUycGP+p/lpDD55A\nmkQ/BpwPHJFSVgyyzYPAuUAHsExKWRNSK0eD2w3XXqscRq9e7pf5ISgdazBAUZF6nHKKKuC1cyfs\n3q2cekaGipYZQfSC9Eps191B3uGtyPJyxAMPkBVvkR9utxoJX3XVMS93FBXB6tWqcmXfJJ7DoUbO\n/rj3XjjvvGPDSe+5Z/DjvvWW2tafQ9++3XdRjjPS09MpK0unrGzC0Rj3/fu3k5rag8dTiNvtJjlO\nkrE0oSEQyeVx4HfAk/5WCiHOBaZIKacJIU4BHgYWhs7EUdBv8nNIZ56eDiedFNpjG40wYYJ6nHYa\nNDQo59Eb415VVqbuFAKYtHLsayLzpqvJA2zX3IblojNCa2ukOHJEXdgGjLpnTZ/uW5+dPXSqv80G\na9fCZz4DDz8Mzc3qonnGGXDNNf6Pu2qVksD8TXo/+aSSWiJQYz0QRlpdMDMzk8zMTHJzs1i1ag9O\nZw/btm2lqCidCRMsmM3muGvSoSstBs+w/2Ep5ftCiIlDbHIRvc5eSrlGCJEjhBgnpTwSKiNHxGCT\nn/4wGmHu3PDZkpICEyeqh9Ppi3Gvq1Pr+2Lc/Tj3Lff8k7mrHwHA/fjTWPLiNwwNh0P9TR3Qt9Nk\nUhe3vl5/Q6X6792rtv3HP9ScSH6+0tvnzFH/5+uvP/491dVw+unHvtbeDjffrO6WVq2KaqRSKGlo\nsGE2T8Rszsfr9dLa2k5Dg42kpDqKizMpLlbOPV7quGuCIxSX7BLgYL/lQ72vRc+hDzX56Y+uLuUQ\nIoHJBFOmUH3wIFVXXKGc+tatvgQmsxkyMpCeHhyXXMlcbzvb809j1p9vJ+5vnvuyIAc4k027d1Mh\nhG+uYahUf69X/Z0xw9e522CAs86CO++Eb3zj2AvBzp0qd6CtDe64Q30fHA51rIsvhscfD+EHHD2j\nqQHu8Xioq7NjNpcDKsY9K8tMVpaZnp4erNY2Dh60kZx8gJKSbMaPt5CTkxOzoY26HnrwhMKh+/s2\nDOpFly1bRnl5OQBms5nKysqj/7S+SZBRLb/8MlWPPQZdXVT3HrOq9++gy1lZkJMTmuMHs7x2rVr+\nwhegvZ3ql1+Gbds42d5D+gP/w3qg4XPf4PJrL1Tb906k9jWDiLflDxsaWAiI3ots3/oClwuADUeO\n0L5+PVVTpkBBgd/zZ6qvV3relCnHrs/NRTY3s/7xx5n/rW8d3b74739nuhDw619T3ZsQdsz/o5/T\niPj/389yTU3NiN//6quvUlvbyVlnVQK+Ql/z51eRlJTErl2bAKisPI1Dh1pYufIfJCe7uOCCsxg3\nzsL69esRQsTM+aipqYnq8aO5XF1dzYoVKwCO+stACCgOvVdy+Ye/SVEhxMPA21LK53uXdwBL/Eku\nEYlDr6hQk5BDaeYDOfVUNSkXA7hvvIXkh5ar5//3Z5I9TiUJ5eYeL1XEGx4PfOUrKhroc5/zvf7a\nayqM8Ykn1F3VySercNDB9pGTAzfeeOxE6E9/CnffreLSK/p9Tb/0JfjPf5TWnuAyw4YNtdjtRWRm\n5gT8Ho/HTVubjZ4eG2lp3ZSV5VJYaCEzxstcjDVCHYcu8D8SB1gJfBt4XgixEGiNqn6+YYPSV++5\nBzZtUpEVHs/Q7+lrURRNHA7IyiIZaLnxJ+T+7i6SpVSOaO9eJcs0Nio5wWKJm1Kyx2A0QmWlCufs\nz+7dKiEoJ0fp6EOl+huNcM45qiF3f+rr1XmZPfvY1995R4UmJrgzd7lcHDnipKAguDkWozGZvLxx\nwDi6u13s2mWjtnY/WVleJk60kJeXS3qcNXMZywz7LRdCPAN8AEwXQhwQQlwlhLhWCPEtACnlv4C9\nQojdwCPADWG1eDiMRqWNfvSRcu5XX40cSiPMzBx8NBhG+sfYHvjT66qqI8D27eT+7i71XAilEy9Y\nAFdcoWLcZ8+G1lblFBsbh79YxRqf+Yy6G+qrh9Pejvv99+Gyy3yp/h99pP6+9Zb/ffzkJ/D222ry\nFFTW7muvqRF6/0iOmho1KX7mmeH9TCFkpLHXTU02hBhdqn9KSir5+eMpLJyDEFPZtg3eeecTPvxw\nK/X1Dbh6pbFIoePQgyeQKJevBLDNjaExJ8TMnAmPPIL4v/9Ty5MmgdWqnEmf9GMwRG5CdADSKzlY\ncR5lW/9NW+FUchpqBx9JGgyqyca4cb4Y99271aSf261CAc3m2O+odNJJKkX/oYdU4tC+few6/3xm\nn3KKL9V/zx71mdxu//s48UQlz1xzjdrHoUPqjuxrX1Prt25V6fwbNqiL4l/+os7T8uUR+pCRZ9++\nZjIzQ1fEzGRKw2QqAUro6upg0yYbUtaSn59MaakFiyVXd0GKQRK+lsvu7yxn6u9v4dAH+ylZWKpG\nh/feqyJhQI1w29oi3iPUurGBwspiAOrvfpziO5aNbEdut4pxr61VjtDrVXcd2dnxJTN4vepzXHFF\n3PVrjTYdHR28++4+CgvDOzCRUtLRYaezswWDoZWCAhOlpRZyc3PjLsY93ghUQ09oh+7u6CY5M5XD\n886iaNObx648fFglpvT9jSDrrnuU+Y+oJhSNmw9TMHdcaHbscikteds2JcnEU5OOlhYls3z2s9G2\nJO7Ys+cAu3Ylk58fuZ6rUkocjja6ulpISmqjqCiDkhIV4x7KJh0ahXbogLX4UxQ2bEK6PQhj9L9k\n0tODY9wUsmz7eTh3Edc2rUaEqzVcZ6eSIrZuVSPfGG7SUb15M1VmM5x/PpSVRducqBJs7LWUkurq\nTaSlzSI5OToSiNfrxW5vxeWyYTQ6KCnJOhrjPpoEJh2H7kNXW9y+ncKGTTT85inGx4AzZ9s2xJw5\nZAEHHnuTmZON4XPmoGSLadPUw25XI/atW5VOLYQKg4wVaaOnZ/BUf82QtLW14XSayM6Onp5tMBjI\nybEAFjweDw0Nrezf30hKyn5KS82MG5dLdnZ2zCYwJRKJO0Lv+/JEu/46IO+4A/GrXwHgbHJgyoti\nWdPWVlVPZutWNXeQlKTC/aIZ4261qgnsxYujZ0Ocsm3bJzQ05GA250fblOPweNy0t7fg8dgwmVxM\nnKhj3EfKmB6hH7p9OSWgHFcUcbV2kZqbjgDkbbch7r2XqFcMMZvVo6JChf3t2+eLcTcaIS8v8jHu\nQ6X6awZlYKp/rGE0JmOxFAKFdHe72L27hdraA2RkeJg40UJBgUXHuIeYOAqDCAzp6qbk3ls4OPOs\nqOqxB598m9Rc9WW1r96EuPfeY9ZHPcZWCOW8TzpJhft98YuqGmGkY9ydTqoD7RQ1Bgjme9HS0kJP\nT3ZcTEKmpKSSl1dEYeFskpKmUVtr4J139rB69Rbq6upxOp3HvSfqv5E4JOFG6C1TF2ABijf+OyrH\nl17JmtJLWFj/Vxw5JWQ07ScrFjT8oTAYVIRJYaFKYrJaVYz7jh3KqaelhS/GvaVF5QfEU4hljHDw\noI309KJomxE0KsY9DSimq6uDrVtb8Hp3YrEYKSuzkJdn0THuIyShNPQ9/9zO5PNns+fnTzH5p18L\n23EGw76nkawphQC8e/kfOeOZ6yJuQ0hxu31NOj75ZNRNOvxSV6c6MI2RRtChwuVysWrVDgoKKhJi\nslFKSWeng44OG0K0UlCQejTGXTfpGKthi1GcCG2492nG3/51AA6vP0TRicURtyGsuFzHNekYdYy7\n3a4mY2OkuUQ8cehQA5s3eygoGKTnahyjEpja6ey0YTC0MX58BsXFueTm5saFvBQOAnXoCXOfu/Pb\nKq370AcRngj1erGXzWb87V+nbsaZyB5vQM487vTB1FSVZn/uuapa4tlnK2d+6JB69DWvCIa2Npg7\nN/7ORRgJ9FyoVP/47xvqDyEEmZk5HDiwH4ulgubmfNaubeOttzazbdsntLS04O2ri685hoTQ0N0d\n3Uz/wy0cnncWJYsiOBG6axdMn04WsO+hf1J+w3mRO3Y0SUtTUSlTpypH3teko65OSTFm8/Ax7l6v\nGtmXlqqRvyZgOjo6aG8XFBZGMfw1QhgMBrKzc8nOzqWnp6c3xr2J5OT9lJbmUFRk0THu/UgIySUa\nGaG7rvwF0578qVpob/dVSxzLtLUpOWbbNjXZmZSkEpj8tXfTqf4jJhqp/rGGx+PBbm/B7bZhMjl7\nE5hUjHsiOvcxo6H3bNlO0rzZKiP0++GfCO22u0jJVg6q4Qs3MP7lh8J+zLhDSuWw9+1Tzt1uV7Ht\nubmqvyqo0Eid6h80sZDqH2u43d20t7fQ02MjPd3NxIm5FBRYyMhInDuYMePQIzkRuumPq6m44TQA\nWv6zntwzTxzxvsZMnQopVU3yPXuUc3c6lSwjpSp/m5w8ds5FAAx3LlpbW/nwwyMUFs6InFFRYt26\naubPrwrqPS6XE7vdhtfbQna2pKwsl/x8C2lpaeExMkKMiUxR64+XUwgRyQhdPeXrLN7zNK2Yyeiw\nkpuuQ6kCQgiVNFRQ4Itxf/99JVO99x5Mn+5rDq0Zlvr6ZlJS8qJtRsySmmoiNbUYKMbp7GTrVhtS\n7sZiSaK0NJe8PAup8d7KcQjidoQuXd0IUyoHZ51N6bY3QrJPf3QfaSGlSEUTvPP5+1nyt1vCdqwx\nw2uvwZHeLoVdXUqOmTlTTbIWFuoko0HweDysWrUFs3nemA3fGyl9Me5StlBYmMqECapJR7zEuCe8\n5GIr/RSWuk30uDwkpYTny23/84tkXfNldbwN+7CcMDEsxxlTdHTAk09CcbHPcbvdSnN3uVQEzezZ\nMHmyar+XgBNcI6WxsZF16+wUFk6Otilxi69Jh0pgKipKZ8IEVcc9lpt0hDQOXQjxWSHEDiHETiHE\nD/ysv1IIYRVCbOh9XD0SowPl0FvbsdRtYs/PnwqPM5cSOX8BWdd8mabpi5A93pA78zEbe33woHLS\n/Ubh1Tt2qJF5aalKVtq0CV58EZ5+WrWRs9liompmJBjqe6FS/ceO3LJuXXUY9irIzMymsLCcvLwK\nWlsLWbu2nVWrtrBly25sNltcx7gPe0kSQhiA3wNnAvXAWiHE36WUOwZs+pyU8qYw2HgcJWepzu7h\nSO+3rd+LZf5kBMBf/0r+xReH/Bhjms2bVemAwUhJgaLe+iROJ6xbB2vWqAiZ2bNh4sSh35+guFwu\njhxxUlCQHW1T4pI331Q9x3fsUCkQ06fDpz5lYNo0M9OmmTGbe7Ba2zh40EZy8gFKSrKPNumIpzDI\nYSUXIcRC4E4p5bm9yz8EpJTynn7bXAnMl1J+Z5h9jVpyqblqOZUrenuEhjiJaPd1/8vUR76vFlpa\ndH2RUGOzwXPPqZF4sHR2qkqQXq8azc+Zo/aTQKFpQ5HIqf6R4J134HvfO/Y1o1GlSLhcqm/5qafC\nlCmQktIX495Camrn0Rj3rKysqDn3UEa5lAAH+y3XASf72e4LQojTgZ3ArVLKuoAsDQJ3RzeVK26h\nfs7ZIXXm7o5uujNzmUondWdfxYQ3/hyyfWv6sXev+hWNhPR0X/apwwHV1UqGKSlRI/eSEqW/Jygq\n1X9StM2IW0pKjn/N4/FVrPjTn9QDYMIEI1OnFjBtWgFvvOFm/HgbM2YcYvbsbk4/PZe5cy1kZcVm\nk45ANHR/V4WBw+yVQLmUshJ4C3hitIb5o2XaAgDG17wWsn1ue2ItyZmpZNDJ1kc/jJgzH3MautcL\nW7ao7kgDqN68Obh9ZWaqX2hJiUpaevNNWLFCRc/s3asaZsQp/r4Xfan+aWlj426kj1Bq6MVB1Mqr\nq1PjhT/9CfbvT+ajj8bxl7/M4q67prNkiYe8vFq+/vXN7Nw5gvpFYSaQ4VId0H84PAGlpR9FStnS\nb/FPwD0MwrJlyygvLwfAbDZTWVl5NJGi78vsb7n5/e1sa9jEe9fewRd70/uH2j6Q5RcXXUDBR68y\nlWSwO2hc98ExiR2j3b9e7rdstVL98cdQWEjVvHlq/QBH3rc8cP2gy1u2+Ja9XqrffRfeeIOqWbNg\n8mSqbTawWKg666zof/4Al2tqao5bX1Y2GYPBctTB9SXbJPpybW1NyPa3cyekpFT3XuvVeqju/Ttw\neTHgRI1Nu4ET8HiceDwfkZSUTE/PGdTWmli9+gPq61PC8n2orq5mxYoVAEf9ZSAEoqEnAbWoSdEG\n4L/A5VLK7f22KZJSHu59fjHwfSnlqX72NXINPZQZoW1tR/Xx3df8iqmP/nD0+9QMzTvvqJrqkehM\n1NOj9PauLiXxzJihYtzHjQtPk44wolP9R0ZtLdx3H6xfP9gWEnChHPfAhwBMxzwyMkxkZaVw442C\nZcv8SzjhJGQaupSyRwhxI/AGSqJ5TEq5XQjxc2CtlPJV4CYhxIWAG7ABy0Zl/QBc9z5IKoQmI3Tl\nSuKYwBYAABQkSURBVLjoIgCa1+xm6slTRr9PzdC43eoXFqk2c0lJqr0eKKF0924l95hMMGuWmvnK\nz4+LBKa2tjacThPZ2WPbmdfVqfSFVavUtbo/t92mygL1jfWEgOZm9cjJ6WHKFCdZWU7ef99JT0+f\n03YBKficdiaQ3/tcucXp0+Hqq9VkqRCqh3msB7zEfGJRX0aotfJsCj8eeUao9EpaT6gid9O7yHkV\niJqPo/qDHlP1S/btU/r2hAl+V1dv3nxURgkrbrf6lbvdKjpm7lxV491iiZlf6sDvxbZtn9DQkIPZ\nnB89o8JAc7P6Srz3nqpC3d7uW9fnoD/+uJrKyipApSb8+MdqEjM3VznbadNUG9yTToKMjG5cLifd\n3f2dthOjsYfsbBOpqSZOP92E3d7nwFPxN4WYmQmXXqoc+aJFMfO1SJxaLn09QvPWjHwitGVzHbkV\npeQCzsefxbTsspDZpwmA7dtjI7wwOdkX4+5yqfvxNWtUXPvcuSrGPYZCVT0eD3V1dszm8mibMiwd\nHbB2rXrs3KmcdF8Eye23w3kDWgXs2AEPPOBbzstTDvrEE+GMM5RjTUvzVaVevBhWrfLS3e2iu1s5\nbimV03a7nbhcSWRnm8jJMZGZacJkMmMymY7pTfrb38K3vuXf/qwsdWO3bRuMj+OqxDE9Qm9fs53s\nhaPrEfrOl37PkpdUeLzrUBOpxWMn0y4m8JfqH2t0dqq8AymVFDN7tirrG+Ua99FM9Xe7VcDQli3K\nQe/erZx0Tw/ceiucc86x22/ZAjfeeOxr+fmqHtvXvqamMQLF43EfHW17PL7RdlKSh6ysVHJyTGRn\nm0hLM2EymUhNTQ2ots1bb0Hv/DigLhheLyxZAjfdBJ/5zMijasNNYtRyGcVEqMfpoTW9mHzZyIfl\nl7Fo77NB70MTAnbsUDFgkZ5FCoTWVt/QrA+HQ02aS6mGarNnK6louA5MYWDDhlrs9iIyM0OTGetw\nqCrGu3b5HnY7XHEFDFT/duyA6/z0OJ8xA777XZg/f3S2SCmPjrZdLt9oG5ykpoqjTjsrSzntvtH2\naBJ79uxRNeAuvFBN50yZoropRmpqZzTEveTyziUPsgTVIzRYV9D6zkbMVZXkAzUPvsui75weBgtH\nx5jR0IdL9SeCGnp/mptVPXYhVObplCnKW02apHT1rCzl7d56S21fVqYmVEtKVH/VMNH3vQgk1V9K\n5aQ/+USNoHfuhMZGNed/0knH6r979ihdeCAGg/o4mQPyZObPV1UXRktPT89Rp+12+5y2EC4yM1PI\ny1MySXp6JiZTPiaT6WiRrFD/RsrKoL5e3TkkKjHp0N0d3Sx5+WZ2Tjyb6UFmhNZ9+btMeFGJc91t\nXVRm+2l/pokMNpvyMCNJ9Q83fZp+T4/qadrQAB99pHR2t1t5udJSNaSbPFmN2vfuVesnTVKvjx+v\nlkOIlOq0VVfb+M9/LLS1Cc44AyorVRRmn5M+fBguG2QqaP58JSP0d+jz5oXGQQ+G2z30pGR+fp9M\nYjkqkxgiLMEZjYntzCFGJZcR9Qh1OI5qnrsu/wnTnrlrNKZqQsH69eoRq7NMl16qvORwmEzKO3Z3\nq9m70lI1gVpWpjznggVqsnUQHbfPSW/ZouaHTzpJvcXtVlMMQigJf9Ixmf1bgElABl//uprM6zMD\n1HWos1PpwJHSfb1e/5OS4CQtbeCkpOm4SUnNyIlbycX6znYKGzapHqEBOvODj75O6Td7mw1v3860\nmTPDaKEmIIZI9Y8Z8vJUgPNwOJ2+51YrWK3IjRuRRiM8+Duk0UhSSTHMn4/7wT/SmZaH06kG8QNj\npkHptsuXq8F9X1BNVpaKsEhNhby8Dj74QFBYOHhkUFJS+OZsh5uULCrqG23nYDKNC3hSUhN+Ys6h\nF1ap0riBNHyWXsnBivMo2/pvOkumkn6gNnYjKQaQ8Bq61aqGnwE49Kho6KDuHAJx6H4QHg/C4wGg\np9sLe/fS4+7B3dZBTlEe2dnwt7+p+OqKCjWoH8rnGQxKolfp3yrVP5wENymZHZJJyWBJ+N9IGIgt\nh/7gg+pvABmh1o0NFFYWUwbU3/04xXcsC6tpmiCprVUaQZSRXklPcyvGun0qwan3IffuRaASwEfq\notymDDwewaoJV/BS/vVMvnAup9bBgvGqT8dIfJGUkv37W8jJmTVCq45lNJOSmvgjZjR0r7MbQ1oq\nnaedTfp7Q2eEvnflo5z+5DcB6NxzmPRJ40Juq2YUuN3w+OMqHiwczsHrVVpGPwd99BEkbowk4wn8\nDSaTEsUXLuS/i27mtF9/DjdKJzYaVXRjV5ca/C9fDp//fHD2tLa28uGHRygsDCJwm+EnJfse/WO3\nIz0pqRk5caeht0xbQB6Q+tbgGaHS04Nj3BROt+1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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3172,7 +3628,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### Matrix multiplication – Projection onto an axis\n", "Matrix multiplication is more complex to visualize, but it is also the most powerful tool in the box.\n", @@ -3184,7 +3643,9 @@ "cell_type": "code", "execution_count": 90, "metadata": { - "collapsed": true + "collapsed": true, + "deletable": true, + "editable": true }, "outputs": [], "source": [ @@ -3193,7 +3654,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Now let's look at the dot product $P \\cdot U$:" ] @@ -3202,7 +3666,9 @@ "cell_type": "code", "execution_count": 91, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3222,7 +3688,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "These are the horizontal coordinates of the vectors in $P$. In other words, we just projected $P$ onto the horizontal axis:" ] @@ -3231,14 +3700,16 @@ "cell_type": "code", "execution_count": 92, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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ps0xMTDg2j3su2Fwu2cv7d7De3l66uroACAQC9PT0cDTZua18mG27tLZXeCUe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\nq1ocHZ4AAAAASUVORK5CYII=\n", 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FLBbj/PlpmptbgPgY9+ZmH+3tN+Lz3UEw6OfFF0McOfI6J0+eZmJiImfzuOeD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\nEaDsbg6r6tdUdaeq3ki8nzi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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3267,7 +3738,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "We can actually project on any other axis by just replacing $U$ with any other unit vector. For example, let's project on the axis that is at a 30° angle above the horizontal axis:" ] @@ -3276,14 +3750,16 @@ "cell_type": "code", "execution_count": 93, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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Yw9///ndeesmZRuvZsxezZ88mLS3Nspi01D/wacpF+UVtbS1lZc7OvbY2ishI\nZwGTjnH/KYfDwVNPPcXbb78FwODBg3nhhRc7N/mVl5SWFjN0aDTZ2YefTldZS8ehB7gOjbFtaIDt\n270eS2ckJibSu3cOY8cOYcyYnvTt2+Aa474Bm62kZbHhIwnlceh2u53777+fE08cydtvv8WYMWNY\nuvRL5sz5R5udub/borXU3+JCn9274bvvfvKQjkP3nObQg8mHHzpHB3z9NcQE1hWwiJCcnExycjJH\nH92Lqqoqdu+2sX37ThobE4mJSSc5OT1sxrg3NDRw55138uWXSwE477zz+J//uS/gFimpqamkW7c4\nYqx+PX31Fdx1l3NIo5cqpMORplyCiTFw4YUweLCzYw8CDoeDyspK1xj3auz20B7jXltby29+8xvW\nrVsLwJQpV3DLLbcE7P/Vp6X+nrr6aoiPh7/9zepIAo7m0EPVnj2Qlwdz58Lo0VZH45Hm5uYDxrjX\n0tyc6prHPfjHuFdUlDN16lS2u1JiN974W6655pqA/n/Z7XaqqtYwYcKQwPjkVFnprJB+8UU491yr\nowko2qEHuE6Nsf3wQ7j9ducEXsnJXo3LX/Yv0rFtm41PP/2CIUPOaBnjHsid4MH27NnDZZddRk1N\nNQD33PN7Lr300g4fz59j8svLS+nRo5oBA472y/nckp/vrJBevZr8NWt0HLqL18ahi8irwHnAHmPM\n0DaeHwd8CPzoeuh/jTEPexiv8sSFFzo79ddfhxtvtDqaDtm/SEfXrl3Zu7eYIUPi2LZtO6WlTYik\nk5SUQXy89SNADmfr1q1ccsnFLduPPPIoZ511loURea6pyUZ2dnerw/ip8ePhyivhs8+gWzerowk6\n7V6hi8gYoAZ47Qgd+nRjzAXtnkyv0L2nsRGioyGIrmbdUV9f71qko5yKCkNERLqrgCne6tAA2LBh\nA1dc0Tqp1LPPPscpp5xiYUQd4/dSf9UpXrtCN8b8V0TaK13TV4S/WT0qwUfi4uLo0SObHj2y2bdv\nH2VlNoqLf6CkJJKIiHTLFulYuXIl06b9umX71Vf/zvHHH+/3OLylqspG//4Z2pmHGG/dej9ZRApF\n5F8iMtBLxwxpOsa21eHaIiEhgV69chgzZghjx/aiX78mGho2eDTGvbOWLFnCiBEntHTmb7/9DgUF\n3/isM/fXOHRnqb/164Yeib5HPOeNQbHfAL2MMftE5BxgHnDs4XaeOnUqubm5AKSlpZGXl9dy42P/\nL1C3w2vp25ZrAAAfO0lEQVR7vyPtn5SUREFBAcYYRo8+gT17bHz88Uc0NcUyYsQ5JCWlUVj4X4CW\nm4r7O8eObP/rX/9ixozbAIiMTOf99+eye/dmKit3AP06ffzDbW/cWOjV47W1PWjQSNLShBUrVgDW\n//4Pt11YWBhQ8fhzOz8/nzlz5gC09JfucGuUiyvl8nFbOfQ29i0CTjDG2Np4TnPovvL99840jAUT\nO1nF4XBQVVXFrl02tm+vwm5PIjbWuUhHR8Z9G2N49913+Mtf/gJAenoGb731Fl27dvV26JYKulL/\nTz+F7GwYGL4f/r0926JwmDy5iHQzxuxxfX8izj8Sh3Tmysc+/BA+/hgWLQqbmewiIiJIS0sjLS2N\nY49tprKykp07bezcWUxzcwrx8c4CpvbyxMYYZs6cySuvzAKgT5+jeeWVV3y/0IMFWkv9B1gdivu2\nbHFWkS5bFrL3jryl3csYEXkL+BI4VkSKReQaEbleRKa5drlURNaIyCrgaeAyH8YbMryeH7ztNmcl\n6VNPefe4fuCNtoiMjCQjI4PBg49hwoTBjBiRTFpaCWVl31JSsoWamioO/nTocDh4/PHHGTlyBK+8\nMou8vGF88cV/ee+99yzrzH2dQw+YUn83tLwurrsOcnLgwQctjScYuDPK5fJ2nn8BeMFrEamOiYyE\nf/zDuRDvWWfBkCFWR2SZA8e4NzU1YbPZ2LZtB6WljUA6cXHJPPbYE3zyyX8AGD/+NP70pz8RHR1t\nbeB+UFe3l8GDLZ6Iy1MiMGuWs0J64kQIwmGi/qKVoqFm9mx4+mnnBF6x/h/eF8gqKio4//wL+O9/\nVwKpnHnmJO699wESE4Oz2tZTAVfq76l58+COO5wV0mE2gZeuWBSupk6FhQuhoCDo5nrxpq1FRcy5\n7z4cO3bQnJXFh2vXsmatc8Ksu+++m/vvv9+1SMcWSkoiiIjIsGyMu79UV5eTk5MSnJ05wEUXwfLl\n8OOPMLTd8RlhSa/QLZLvy/USjQmqClJvt8XWoiKeO+MMHty8mUSgFpgCDLjrLh597LFDbpLW1tZS\nWmpj69Zy9u2LJioqg+TkdEsW6fDlXC4lJRs55ZTuQXOz16fvkSCjV+jhLIg6c194/rbbWjpzgETg\nTeAvO3a0OeIlMTGxZaGO6upqSkrK2bZtPeXlcURHOzv3QJvH3FONjQ3Ex9eTkpJidSjKh/QKXYWM\n9evXM3DgQMYAX7Tx/IzTTuPBRYvcOpYxhsrKSvbsKWfbtkqamhKJiXGOcQ/GlEVZ2S7697eTm9vT\n6lBUB+gVugobBQUFjBw5smV70IQJ1C5axIFzNdYCEdnZbh9TRFrGuPfr56CiosK1SMc2mpuDb5EO\nZ6l/H6vDUD4WHK/GEOT1cehH8s9/QkmJ/87noY62xaJFixCRls582bJlGGP4/SuvMKNvX2pd+zmA\n97t1Y+pDD3XoPBEREWRkZDBokHOM+8iRqWRklGKzrXaNca88ZIx7R/liHHpdXS1paRIQC1J7wq3X\nxaOPOguPFKBX6OFh5Up46y344IOQyK/PmzePn//85y3ba9asYdCgQS3bvfv04XcLF/KX++7DsXMn\nPWNiuKaggMi4uE6fOyoqiszMTDIzMxk4sMm1SMcuSku3AOkti3QEkpqavQwdGtgTcXVYZKRzZNei\nRRAkn5Z8SXPo4aChwVlwdOutcM01VkfTYbNnz+baa68FnNPsrl+/3v2Ji2bOhHHj4LjjfBJbQ0MD\nNls5xcU2bDY7EREZrnncE3xyPncZYygrW82ECQOCojrUY83NzkUxLroIpk+3Ohqf0SXo1E999x1M\nmOAsOOoTPLlUYwxPPfUU011v1pycHAoKCugWwKvZ1NXVuca426iqgshIZ+ceG9v5Twieqq6uIDV1\nD8OG9ff7uf3mxx9h1ChYvNi5gHoIcrdD188oFvFrDh2cUwHcfbdzZfXmZv+eux1ttYUxhnvvvZeI\niAimT5/O0KFDXemNbQHdmQPEx8eTk5PN6NGDOfXUPvTv78Bu30RJyTr27t1NU1PjYX/W2zn0urq9\n9OwZZKX+Lm6/R44+Gh57DK64wvlpNIxpDj2c3HYbbNwIu3dDjx5WR9Om5uZmbrzxRl5++WUAJkyY\nwMcff0xCgrWpi47aP8a9V68e1NTUUFJio7h4PeXlsURFZZCSkk5UlG/mkLHb7URFVZOenuuT4weU\na6+FPXugtjasp7zQlIsKCI2NjUyZMoW5c+cC8Mtf/pI33ngjJCfMMsZQVVXF7t02tm+vpLExkZiY\ndJKT0706xr28vJQePaoZMOBorx1TWUPHoaugsG/fPiZOnNjy8fqGG27g+eef933xzh13wKRJMGKE\nb8/TBhEhNTWV1NRU+vVzUFlZ6Rrjvh273Xtj3JuabGRnd/dS1CoYaA7dIn7PoQeYiooKhgwZQmJi\nIvn5+dx77704HA5eeukl/1RinnACXHkl1NX5/lxHEBERQXp6OgMH9mXChCE0Nq6hS5cyystXU1JS\n1OEx7qFQ6h/u75GO0Ct05Ve7d+9m+PDh7Nq1C4Ann3ySvLw8TjvtNP8GMnmyc5Wn3//eOd1wAIiM\njCQ1NZWhQ/sxcKDddRN4NyUlWzAmrWWMe3srMAFUVdno3z/DrX1V6NAcejhzOODee52dmo+v5IqK\nijjuuONobHSO8Jg9ezZTp0716TnbZbPB8cc755D/2c+sjeUIGhsbsdnK2bbNRllZEyLpJCVlEB9/\n+MrPkpI1nHpqn6CrDvUau935R/vFFyEE1oTVYYuqfRERUFbmLDjykbVr1yIiHH300TQ2NvLBBx9g\njLG+MwfIyIBXX3WOkKiosDqaw4qJiaF7926MHDmA0047liFDIomM3EJJyRrKynZQX//TtFGwlvp7\nVVSUczjjtGnO6aTDhHboFgmY/OCTT8LnnzvTD160fPlyRITBrkKPxYsXY4zhoosuOmRfS9vizDPh\nz392dgABoL22iIuLo0ePbE4+eRDjxh3NgAEGY36gpGQdZWW7aGxsoKZmL717B3+pf6dfF3/8o7Po\n6B//8Eo8wcCdRaJfFZE9IrL6CPs8KyLfi0ihiOR5N0TlU8nJ8NprcMMNznG8nbRw4UJEhJNOOgmA\nFStWYIwJ7IUKJk0KyiXNEhIS6NUrhzFjhjB2bC/69Wuivn49sbGl2O12mpqarA7RWrGx8PrrcOed\nYTOBV7s5dBEZA9QArxljDln3SUTOAW4yxkwUkVHAM8aYkw5zLM2hB6o//AHWrHFeqXfgRtrcuXP5\nxS9+0bK9fv16jvPRvCnq8MrLy1m06EcSErogUkH37gnk5GSQlpYW9It0dNgTT8D8+c4JvIJwLnvw\n4jh0Y8x/RaT3EXa5EHjNte9yEUkVkW7GmM5f7in/eeABeOQRaGz0qNJu1qxZTJs2DYDk5GTWrFlD\nr169fBSkas+uXTbS0nqTlpaJw+GgoqKKXbtsREZuJzs7iexsZ+ceLPO4e8Xtt8NRR1kdhV9447fa\nA9h2wPYO12PqCAImh75fTAw8+KBbnbkxhscffxwRYdq0aeTm5lJSUkJVVVWHOvOAawsLP0V2pi3s\ndjvbt1eTnJwOOMe4JyenkZV1NGlpQygpyWD5chuLFq1m/fofqaio8No87r7gtddFZKRznpcgvTr3\nhN8/g02dOrVlytO0tDTy8vJa8qv7f4G6HZjbixcvZubMmbz77rsAHHvssTz55JNMnDixU8ffz+r/\nX35+PjgcjH/wQZg1i/ydO/1+/sLCwg7//Pz589m4cR8/+5nzNtb+ib5GjBhPZGQk33/vvA2WlzeG\nHTvK+eijj4mObuD8839Gt24ZfPPNN4hIwLzeCgsLLT2/ldv5+fnMmTMHwP0ponFzHLor5fLxYXLo\nfwMWG2PedW1vAMa1lXLRHHpw2FpUxJz77sOxYwcRPXpw5QMP8PCjjzJ79mwAzjzzTObNm0d8fLzF\nkfrICy84bxQvXRowo1/csXLlRqqru5OUlOr2z9jtTVRW2mhuthEf30ivXulkZWWQFIQ3iUOZV+dD\nF5FcnB36kDaeOxf4reum6EnA03pTNHhtLSriuTPO4MHNm0nEuRbnFOBD4PLLL2fOnDkhOWHWTxgD\nZ58No0fD/fdbHY1bGhoaWLRoA127Du1wdWhjYwNVVTYcDhvJyQ56986gS5f0oJ3p8oiMCarVu7xW\nWCQibwFfAseKSLGIXCMi14vINABjzP8BRSLyAzATuLGTsYeFg9MNgWLOffe1dOYAicCbwIzLL+fN\nN9/0SWcecG0hAn//Ozz/PKxY4ddTd7QtyspsiHSu1D8mJpbMzKPIyhqEyDGsWweff76Zr75ay86d\nu2jw81zjPntdlJQ4F8SoqvLN8S3kziiXy93Y5ybvhKOsZLPZ+PJ//5eD6wsTAeOaeyVs9OgBzz7r\nnMDr228Dfo7tLVv2kpTkvZWo4uLiiYvrAfSgrq6W1attGLORzMxoevbMICMjPXiXtMvKgqFDnesD\nvPqq1dF4lc7loti5cyd5eXmUlpZyDFAIP+nUa4G/TJnCjDfesCZAK33xBYwZE9Afz2tra1myZAtZ\nWYPa37kTjDHU1lazb185EREVdO0aR8+eGaSnpwffGPfqasjLc1ZKX3ih1dG0S9cUVe3avHkz/fr1\naxm69sYbbzDmlFMOyaHP6NuX3y1cSO8gWos0nPz4YzHffx9NZqb/xlobY6ipqaSurpzIyEq6d0+k\nRw/nGHe/TH/sDf/9L/ziF85PYFlZVkdzRNqhB7j8/PyW4Ur+tnr1ao4//viW7fnz57cMPYQDRrns\n3ElEdjZTH3rIp525lW0RaDxtC2MM+fmriY8fQHS0NSkQh8NBdXUFDQ02oqJq6NEjmaOOyiA1tXOL\ndPjldXHPPbBpE/zv//r2PJ2kKxapQyxdupQxY8a0bC9ZsoSxY8cesl/vPn3CM70ShCorK6mvjyMl\nxbp8dkREBKmpGUAGdrudXbsq2Lq1lJiYrfTsmUa3bumkpKQE5tzsDz4I331ndRReo1foYWDBggWc\ne+65LdurVq0iL0/nUOuQ+nqIi7M6ihbr1m1m165U0tIyrQ7lEHZ7E1VV5djtNuLiGujdW8e4d5Sm\nXBTvvPMOkydPbtnetGkT/fr1szCiIGezwfDhzoKjHtbPbmG321m0aA1paUMCPm/d2NhAdXU5zc02\nEhPt9O6dQdeuGaE5xt0HdIGLAOfLsdcvvfQSIsLkyZPJyMhg27ZtGGMCtjMPuHHoh5OR4VwM49pr\nfTbfiydtUV5eTnNzSsB35uAc496lS3eysgYSGdmPjRsj+PzzH1m6dA3bt++kvr7+kJ8JmtdFANEO\nPUQYY3j44YcREW688Ub69etHWVkZe/fuJScnx+rwQscf/uBc3ejFF62OhG3bbCQkdLE6DI/FxcXT\npUs2WVmDMaYPa9c6WLx4E8uXr2PXrt0tyxRaxuKFwztDUy5BzuFwMH36dJ52LXQ8atQoFi5cSHJy\nssWRhbBNm+CUU5ypl/79LQnBG6X+gcQYw759NdTW2hCpoGvX2JYx7n6damL5cueydV9/HVDFZJpD\nD3F2u51rr72W119/HYCJEyfy/vvvExtAL8KQ9tJL8M9/wuLFlpx+x45dfPedna5de1pyfl9yFjBV\nsW+fjYiISo46KpHs7HTS09N9n14yBi65BI45xrk0YYDQHHqA62h+sL6+nnPOOYfo6Ghef/11rr76\napqampg/f37QduZBmSu94QafrFXpbls4S/2Df93QtogISUmpFBdvJSNjKHv3ZrJiRSWfffYd69Zt\npry8HIfD4auTw8yZ8MYbsGSJb87hQzoOPUhUV1dz+umns8I1WdTtt9/OE088EV4rzwQSEbBoZaba\n2lqqqoSsrINn3Qk9ERERpKSkk5KSTnNzs2uMexnR0Vvp2TOV7t0zvD/GvWtXmDULrr7aWUWakuK9\nY/uYplwCXFlZGSeddBKbN28G4JFHHuH3v/99SORNVcdYUeofaOx2O9XV5TQ12YiLq3cVMDnHuHvt\nvXH99ZCY6JzvxWKaQw9y27dvZ/DgwVRWVgLOoYg33HCDxVEpqwVCqX+gaWpqpKrKOcY9IaGJ3r3T\n6do1g8TETn6CqamBpiZIT/dOoJ2gOfQAd7hc6aZNmxARevbsSWVlJW+//TbGmJDuzIMyh34wY8C1\nZF1ntNcW+0v9w6Ez37+EXnuio2Po0qUbWVkDiI4+lo0bI1myZAtLl65h27Yd1HV0GGJSUkB05p7Q\nHHqAWLVqFcOHD2/ZXrBgAWeffbaFESmPrF4NEyc6/83w3c3KnTv3EhMTfGPP/SU2No7Y2Gwgm/r6\nfaxda8OYH8jIiKRnz3S6dMkI2sED7tCUi8WWLFnCuHHjWraXLl3KKaecYmFEqsNuvRV274Z33vHJ\n4YOp1D/Q7B/jbkw5WVmx5OQ4F+kIluUUNYce4ObPn8/555/fsr169WqGDDlkyVYVTOrq4IQT4L77\n4IA5dLyltLSUgoJqsrKO9vqxw0XrIh3OAqbu3RPIyXHO497uIh2NjVBeDt26+SfYA3g1hy4iZ4vI\nBhHZJCJ3t/H8OBGpEJGVrq//6UjQ4eD1119HRDj//POJiorihx9+wBgT1p15SOTQAeLj4fXX4ZZb\nYPv2Dh3iSG0RrKX+HeVuDt0TzjHuKWRl5dKly1AqKrJYsaKKRYvWsGbND9hstsOPcX/rLWfRUXOz\n1+PyFncWiY4AngfOAgYBk0XkuDZ2XWKMGe76etjLcQY1YwzPPPMMIsJVV11Ft27dmDt3Lk1NTfTt\n29fq8JQ3nXAC3HwzPPqoVw/b0NDAnj31JCYGz5joQBcREUFychpZWUeTljaEkpIMli+3sWjRatav\n/5GKigp+klG46iqIjoYnnrAu6Ha0m3IRkZOAGcaYc1zb9wDGGPP4AfuMA+4wxpx/mMPs3y+sUi7G\nGB544AH++Mc/AjBw4EC++OILMnx400wFALvd+eXFedNDudQ/0LSOcS8nNnZfyxj35ORkpLgYRoyA\nhQuda5L6iTdXLOoBbDtgeztwYhv7nSwihcAO4E5jzDq3Ig1BDoeD3/3ud7zompFv7NixLFiwoPPj\nYlVwiIpyfnmRs9Rf13T1h6ioKNLTuwJdsdubKCqy8cMPO4iPb6RXr3SyH3mEuCuugIKCgFrsBLw3\nDv0boJcxJg9nemael44bVJqampg8eTKRkZG8+OKL/PznP6ehoYElS5Yc0pmHTN7YC7QtWrXVFvtL\n/ePjw+uCwBc5dE9FRUX/ZIz7mjV2/t11OGVdj6L+lVesDu8Q7lxG7AAOnLQix/VYC2NMzQHfLxCR\nF0UkwxhjO/hgU6dOJTc3F4C0tDTy8vJaFoLd/2IOtu1Ro0ZxwQUX8OmnnwJw3XXXMXPmTL744gu+\n/PJLy+ML9O39AiUeK7cLCwsPeb5Xr6OJiMho6eBGjHA+H+rbGzcWWnL+vLzRNDTUs2LFZzQ3NzJ0\n6DCgnjVrlpGQEM1pp51Kw6y/sfLHzcQcsJC1N18P+fn5zJkzB6Clv3SHOzn0SGAjcDqwC/gamGyM\nWX/APt2MMXtc358I/NMYc0gUoZZDr6ys5LTTTmPVqlUA3H333fzpT3/SeVbUT1VWwq5dcFxbYwmO\nTEv9fcMYQ2NjA42N9TQ01GNMPeD8io0VUlPjSEmJIzk5jrg451dMTIxl722v5dCNMc0ichPwCc4U\nzavGmPUicr3zafMycKmI/AZoAuqAyzoXfmArKSlh5MiRFBcXA/D4449z5513akeu2pafD3feCatW\nOSd78sD+Uv+UFO3MO6K5ubml025qau20RRpISoqhS5c4UlPjSEhIIi4uk7i4uPbHowcwLSzywNat\nWxk0aBC1tbUAzJo1i1/96lcdOlb+AR/Vwl1YtMWVVzqnYX3hhSPudnBbrFu3mV27UklLy/RxgIGn\noCC/JR3SnqamRhoa6mlsrKe5ubXjjopqJiUlruUrPt55tR0bGxtUU097c5RL2Fu/fj0DBw5s2Z47\ndy6XXHKJhRGpoPPcc3D88XD++eDmHD12u53t26tJS8v1bWxBwuFwtKRJGht/miaJj48kJcV5tZ2U\nFEdcXFpLmiSc6BX6ERQUFDBy5MiW7YULF/Kzn/3MwohUUFu0yFmc8u230KX9is9wLfW325tarrbt\n9tZOOzLSTnJybEt++8Cr7VCf20bncumERYsWcfrpp7dsL1u2jFGjRlkYkQoZt98OAweCG6m6lSs3\nUl3dnaSkVD8E5l/BdlPSatqhd8AHH3zAxRdf3LK9du3an6RavCks8sZuCqu2cDjgCLnb/W3R0NDA\nokUb6Np1aFB3Yu3dlNyfJklIaO2499+UDKvXRTs0h+6B2bNnc+211wIQFxfH+vXrPRr7qZTb3LwR\nV1ZmQyQjaDrz9m5KZmbuT5NkBOVNyWARtlfoxhiefPJJ7rjjDgBycnIoKCigmwVTYyp1sKVL12BM\nn4CqDvXspmRcWN6U9BW9Qj8MYwz33ns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6ha48Vl9fz969FWzbZqO6GiIjnZ17bKx1f3yChTGGv//977z4ojON1rt3H159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\nhO0VujGGJ554gttuuw2AnJwcCgoK6GH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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3299,14 +3775,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Good! Remember that the dot product of a unit vector and a matrix basically performs a projection on an axis and gives us the coordinates of the resulting points on that axis." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### Matrix multiplication – Rotation\n", "Now let's create a $2 \\times 2$ matrix $V$ containing two unit vectors that make 30° and 120° angles with the horizontal axis:\n", @@ -3318,7 +3800,9 @@ "cell_type": "code", "execution_count": 94, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3344,7 +3828,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Let's look at the product $VP$:" ] @@ -3353,7 +3840,9 @@ "cell_type": "code", "execution_count": 95, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3374,7 +3863,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "The first row is equal to $V_{1,*} P$, which is the coordinates of the projection of $P$ onto the 30° axis, as we have seen above. The second row is $V_{2,*} P$, which is the coordinates of the projection of $P$ onto the 120° axis. So basically we obtained the coordinates of $P$ after rotating the horizontal and vertical axes by 30° (or equivalently after rotating the polygon by -30° around the origin)! Let's plot $VP$ to see this:" ] @@ -3383,14 +3875,16 @@ "cell_type": "code", "execution_count": 96, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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ojB0bePsDgFKpictBVyxpQogyAI9IKd9WYt/BRiu5p0rYaTbTBOTTT1MygNJI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\neCdPBh5/nLarq8lx2i9Q5s8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E+UKIcgCzAXwhhFCFti+ltAC4BZQVtBPAB1LK\nHme4Q4EQYjmAnwGMFkKUCSH+GGqb3GHLCLkKlB2yzfadXBhqu9yQDWCtTTPfAOAzKeV3IbZJy2QC\n+NHpeH4upVwTYpvc8RcA7wshikDZLD2mpHLREMMwTD9Atak5DMMwjPewM2cYhukHsDNnGIbpB7Az\nZxiG6QewM2cYhukHsDNnGIbpB7AzZxiG6QewM2cYhukH/H82eNF0wWpgOgAAAABJRU5ErkJggg==\n", 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04O0PAEqlJq4AXbGkCiFKADwkpXxLiX0HG63kniphp8VCE5BPPUXJAEojJXDN\nNQasX5+H1at7kXZYWuoU73NySMMMYAvXDsdy7FjSzrdtI4euomjVUFjYbT58S0sLLBYdwsO9+1mH\nh4fbI/gkmM0mlJY2oaSkHAkJERgwIB7x8fE+Nf3qyc5eYbUCO3ZQpL1uXeeiMIC+vKeeSrexY7uO\nqCdNoiphi4XmSmJikLdoEWW2WK3AWWdpOqtJEWcupbxCif0wweVf/wLS0oDzz1d+3yUltKzb1KnU\nIiUtzYsnWSzA3Ln0owWoy9eEIJctCEGX2I2N9CY09ONuaGhGeLhvZbtRUTpERekgZQpMplbs329E\nWFgdUlNjkJISD71eH1gZRkqgvNzptCsqPI+bPJmc9syZvT/Bjx5N37G0NJLu4uOBjAzKflqwgLKi\nNAxXgPZTiosp3XbLFuqOqCQ+pR2+/DKweDH9/9JLwE03KWtUbzGbKWWxsdFz0ZHKsNls2L79EGJi\nhijWQtdms6K1tRkWSyMiI9uRnhGH5KQE6HR+9l539NRZt44ykTyRk0PyyCmnAJmZ/r2eAymBQ4ec\n8yrl5fQjmDPH9z49QYBb4DLdcuONwE8/KVsAabORLv7KK8CKFV52OywsdEbfeXnAt98q0/NECZqa\ngI8/pst2lTdSb2pqQlFRIxITBwZk/xZLO1pajJDSCL1eID09HgkJ8Yjo7rOyWOjqat06irg9LZkX\nG+uUSEaNCs48RXs7cPgwXTaecIJqJjq7ghtt+Uh/0MyLioAPPgD27FHOnqYmWpUoN5e60Dq6HXZp\nZ0sL/XhLS2m7pIQ06hDQpY1xcaSjfvghnWBCvPRad1p0XZ0RERGB64wWERGJhIQUACTDHDxoBFCC\nlJRopKXGQV9bi7AffwTWrYOhpuZYyl8Hpk8np33CCaGLhJubqXPmaafBUFWFPJU78t7AzrwfsmQJ\nyR9KqQeOtMPcXArCepxMveceZ2bKJ5/Qk9VKairpqV98odqURavVitraVuj16YF/saYm6H4rgG7L\nVtgOHUAjbKhDGyJgxAC0Ihk2Ok7z5wOzZ6tLoqqro6j8/PPJxqqqUFukKCyz9DN++AG47jpKEFEi\nOHKkHT7+OHDDDT3o42vXOnPEFy4E3nxTVdki3bJrF+W4Dx6sustyo9GIvXuNykoslnZgTxGwdQtQ\n0EXqnz4OmDIVmDIF1uyBaGltgs1mREyMREYGyTCRajn5VVbShKcGJzpZZmE6ISVF5E8+6b8jt1ho\nXx99RN1BRRBOAAAgAElEQVQOp0/vZvCRI84ILTqaMhWSPBe1qJYxY6jd7tatqktZJIkl3rcnSwmU\nl9H72rIFMHbR0mDCRJopHzXa46VXOID4+GQAyTCbTTh0qBFAGZKSIjFgQAL0er2ia5t6jdVK+rgG\nJjr9hZ25G31ZM//sM7rCvOQS/167pAT4f/+PEgO6TTuUEob8fOQ5Ug1//hmYNcu/Fw8AXh/LGTMo\nE+PgQer3EWQ8aeZWqxV1dW3Q6zN63sHRo9TbfesWoLSL6sicXGDqFGDiJJ8nfbeV7MWMEeMhZRpa\nWpqxb58RYWFHgl9t2sNEp1Z+697Czryf0NZGKwi99ZZ/QaUj7XDJEkr06JKVK2Fv9gI88giteKF1\nwsIo46apCaipUYUe3NzcDJstBmFhLo7KbCZZaOtWYEcX6UoJSRRpT5lCs9UBcK4hrTZ1THTOnUsT\n7f0A1sz7CU8+SYHxF1/49nyv0w4PHACGD6f/x4whh+JvXrLaaG6mM5kQodVfbTYcXPcrjL+WIHb7\nTqClyfO4yaRr4/jjVZH2aTab0NbWBCkb/a429YhjolPjFZ0OOM+cOUZ1NRW//forVS73looK4M9/\npt/Ihx92schyeztJKJs30/aePeQ8+iqOg5GYGJyUxZoaZ3XkoUMAACuA7UiAHmnUy3rYcJqQnDiB\nJidVjpQSJlMrTCYjwsKalak21fBEZ1ewM/cRrehovbHzhhsoOH7hhd6/TlER9VQZMYKWk/N4VfzM\nM8Cdd9L/b7zRof2iFo6nzzaWl9NC0ZmZyjW3aW0F/vtfctqOE6PDTthbtmZkAKecgqNTpuD31hQk\nJqkr+tz4e6G99a73+F1t6sNEpxa+mwBnszB2Nm4E/vlP6rnfWxxph08/3UXa4ebNzjSWs86i8neN\ndRr0i+xschzff0+LIvRGJrBaKT/UUR3paR3T8HBndeSECaSDu0yA1h4oR5TNxywWFWEwADZbOLZt\nS8CgQQkYPLgdqWlGpKVWIjHRi2pTs5kuHzVS0RkoODLvw0hJrS2uuoqcsbe4ph1++KGHtEOjkSKg\nujrarqhQrn+GFtm4kU5sXaUsVlQ4JZKyMs/7mDCBJldnzqTK0x6wWCzYvr0EcfG5CBPaPoG+9x6w\neYtzOyqSDiP1xGrFlVcakZDQRNWmaSTDHFvb1DHROWdOn53o5MicwSefAPX1pHd7S1UVrci2f7+H\ntEMpqRnWK6/Q9qpVVOnX35k+nU5wO3aQnr1uHaUAeiI729lj24/0xqamJkip17wjB4CUlI7bZpeV\n7aqqY/DsczGI0ycjI6MeSUlVsNnCcNNNA5ER2dKxorOfw87cDa3oaD3ZaTaTjP3qq95fdTrSDl9/\nnVSTDs9btQo480z6/6abgBdf9CqdTQvHs9c2WiykDaxYQTeTqfOY6GinRDJ6tCKX/q555rW1TdDp\nkv3eZyDoTjO3WOi819jobBtvsXQYAcDc6dbULGAqiUJJSSKmT49CxJE6IDeRWkH4ONGphe9mb2Bn\n3kd58UVqfHX66T2PdU07fP99t7TDigpnBJmWRqmH8drXab1CSsrKWbmSnPb+/Z7HLVhAOfVz5wLf\nfUfPC2CFa3t7Oxoa2pGQENrGXw6kpPNZYyPVU+2uBqzVVKPkcNrTp9PCQL//Tmsxx8dboNebsX+/\nGVabq+MWAKLsNx2AeOiiopCYGI7zzgPm5Fmhb+gfFZ29hTXzPojRCPzP/wCLFlGqd3c0NJCmXlvr\nlnZoswFnn00ROUCa8NSpAbU7pNTW0iTBihXOxTHcGT+enPZll5Ez8URdHe0nISFgKYv19Q04cMCM\nxMTANtaySfp+tLZQ0ZnDMR89SqUEe/bQHK7RSOPj44HaOiAhHjh+FJCUSE47NtaMjAwzoqLM0OnI\naUdGCuj1UdDro/DKK1HYVuBw4HQFk5xEgcj48ZQaP2YMINr750Qna+b9mMcfpw6zPTnyoiKaM7rx\nRqqBOZZd9+ab1I0LoFSWJUsCam9QMZspenZIJJ4CjJQUctpXXEEl/L3J0ElJIY3q00/JCwWgwpEk\nlpSeB3aBTQItzeSYHZJHaip1Iy4qct7X3AxY7S3Ic4eQs46Pp4uOhATg5JPplpAARERYYLGY0d5u\nhs0l0nZ12jExOkRGUaTtWiA0dCiwzT7FMHECXejMmOFW39QPKzp7CztzN7Sio3VlZ3ExLQfX06IT\njrTD558Hbr7ZLn/v2eNczPaEE2j1Cj+73oXseEpJC184nLajb7o7554Lw8SJyLv7buUi6YEDgdNO\no4U2epuy2A2GwkKcNGoUjh5tR0JCTKfHXfXomBjqb7ZvX0enXVVNY2NjyDEnJNBt3DgyNSXFeV98\nfOeCUavVgvb2jk7bZDLDZiOnnZYWhc2HDmDOxImdnHZX5OYC4WFUpezRT9fV0ZtTeKJTK791b2Fn\n3se46y6an/RYpQn6Tdx9N0kqxxZZNplosVvHahX799NK5VqhqorOTitWUJmrJ6ZOpUj7kkvIa7li\nMCgviRx/PHnQjRv96rIoJaWgWyxU3PjV/iYcPBiHtjYBoxEo3EHj9LEkhej15IinTrUrPXo6tzic\nc0wMZT72dI52OO3WVs+Rdlpa15F2XE0lYmM6n2y6Ij+fNHWPGZmOik4/Jjr7C6yZ9yE2bAAuvBDY\nu5d+1O7U1VFwExVFc3ppaaAmWA8/TANWriQ9WK20tdEZaMUKct6eyMwkp3355eTRQtmq1majHu77\n9nk8uzY2UtFifT1QVw/U1QIbNgJlpdR2oaGBHhOCAv2hQ4FffimBXj8AqakxSEykx2JjKWCN1QNh\nvXy7niLtzvJIFCKjoryOtBXBUdE5dCh5+3480cnl/P0Mq5Uuid2q6Y+xZg09dvLJJK+Eb/iZVoIB\nyIGvWKGeHt1SUpK7QyLpakWYiy8mxz1/vip+7FLS5GBlJc3TVVQAv65vR8zarzBqQC1KWtOPOeiR\nI0naMBiA5GS6paRQBB4eThdKyclAcgpJIgBgNptRWFiOhITcXvcuUa3T9gRXdHaAJ0B9RCs6mrud\ny5fTVegf/9hxXKduh5PqgdhM+sEA1MCpy4bkytvZibIyWpB05Upy4J448URy2hdfTH1Jgmyj1UqH\nqaKC6oK2baNAu7SUmphVVgJDhpBzfuQRujjIzCRTjcZItCWejtNHfoKJCfWIH5yM5GS6sNiwAZg8\nmSaqR42ihBq9vuuPY9WmTRgUPbJbR96T0+5OHlGK7tYq7ZEgTnRq5bfuLezM+wBNTcADDwDffNMx\niKmvpyKg2lpg8yaJ7AeuBeYspwfXrqXy8WDR0gJ8+SWdUT77zPOYnBynRDJ+fMCvFFpbqWDzxx/p\n/4oKurIvLyeZdvJk4Lbb6PglJZGDbmykxTluuomk98mTSeLIzaVmZHfc4emVYoH6MyllMa4Zthg9\nVq8GJIAdO52jEhPIjuhomod2OPqcHPpcjza04rihlOOvBqetOAGa6OwvsMzSB7j/firGeP99533f\nfQdcein5xedO/QSRl1xAD9x9N6UNBAqbjUJOh0RSX995TFiY02nPnato+p5D6mhocEodlZU0xztx\nIvDoo7RdWUmO03GBMncuOeu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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3405,14 +3899,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Matrix $V$ is called a **rotation matrix**." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "### Matrix multiplication – Other linear transformations\n", "More generally, any linear transformation $f$ that maps n-dimensional vectors to m-dimensional vectors can be represented as an $m \\times n$ matrix. For example, say $\\textbf{u}$ is a 3-dimensional vector:\n", @@ -3449,14 +3949,16 @@ "execution_count": 97, "metadata": { "collapsed": false, + "deletable": true, + "editable": true, "scrolled": true }, "outputs": [ { "data": { - "image/png": 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ifbFrl3w8fejDD34gy8f/6Z8kbgmHjec4flwuzPp8xvzvn/5U+uTdd6WPdu+W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\nImvkbXENERGZL+cNirnwRii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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3475,7 +3977,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Let's look at how this transformation affects the **unit square**: " ] @@ -3484,14 +3989,16 @@ "cell_type": "code", "execution_count": 98, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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LOGPn4MwZrTN7to4WqqpUgrniCo1CGTpUJZXsbHX4oF8ES5ao7NTU1HZDrlih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\nqIlHknZmztTEHUvaMYwk0MS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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3510,7 +4017,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Now let's look at a **squeeze mapping**:" ] @@ -3519,14 +4029,16 @@ "cell_type": "code", "execution_count": 99, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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SzZoOkHvvtTtCR5Guj40snc6XTQYHzS/O+Hh+5F1XZ0Zy60je5Wzzi8X62bUr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\n/X6eeOIJXve6/Wv5QZVdJr4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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3545,7 +4057,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "The effect on the unit square is:" ] @@ -3554,14 +4069,16 @@ "cell_type": "code", "execution_count": 100, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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f8iIi2UydvEiSqZPvAHXyoaS8kzez6Wa2xszKzeyOBLdfb2bL4n8WmNmkzoQR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\nRUSymIa8iEgW05AXEcliGvIiIlns/wOtCYLlY0szJAAAAABJRU5ErkJggg==\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3576,7 +4093,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Let's show a last one: reflection through the horizontal axis:" ] @@ -3585,14 +4105,16 @@ "cell_type": "code", "execution_count": 101, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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hMhno65vuYU9Omk0LQiEz4sKiadk5FQiYG5jr1l38/MTE9DDCrHe/2/xxeuop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\nAABJRU5ErkJggg==\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3611,7 +4133,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Matrix inverse\n", "Now that we understand that a matrix can represent any linear transformation, a natural question is: can we find a transformation matrix that reverses the effect of a given transformation matrix $F$? The answer is yes… sometimes! When it exists, such a matrix is called the **inverse** of $F$, and it is noted $F^{-1}$.\n", @@ -3623,14 +4148,16 @@ "cell_type": "code", "execution_count": 102, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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FaNMllMxKJwlXvxyf994Dq/WYwuxTb5Vy7YdC8X/uDRvI/u1vOWtwkLff3odS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\n09XVza5dLbjd2Tgc5Vgs0U+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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3652,7 +4179,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "We applied a shear mapping on $P$, just like we did before, but then we applied a second transformation to the result, and *lo and behold* this had the effect of coming back to the original $P$ (we plotted the original $P$'s outline to double check). The second transformation is the inverse of the first one.\n", "\n", @@ -3663,7 +4193,9 @@ "cell_type": "code", "execution_count": 103, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3685,7 +4217,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Only square matrices can be inversed. This makes sense when you think about it: if you have a transformation that reduces the number of dimensions, then some information is lost and there is no way that you can get it back. For example say you use a $2 \\times 3$ matrix to project a 3D object onto a plane. The result may look like this:" ] @@ -3694,14 +4229,16 @@ "cell_type": "code", "execution_count": 104, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3718,7 +4255,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Looking at this image, it is impossible to tell whether this is the projection of a cube or the projection of a narrow rectangular object. Some information has been lost in the projection.\n", "\n", @@ -3729,14 +4269,16 @@ "cell_type": "code", "execution_count": 105, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3755,7 +4297,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "This transformation matrix performs a projection onto the horizontal axis. Our polygon gets entirely flattened out so some information is entirely lost and it is impossible to go back to the original polygon using a linear transformation. In other words, $F_{project}$ has no inverse. Such a square matrix that cannot be inversed is called a **singular matrix** (aka degenerate matrix). If we ask NumPy to calculate its inverse, it raises an exception:" ] @@ -3764,7 +4309,9 @@ "cell_type": "code", "execution_count": 106, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3784,7 +4331,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Here is another example of a singular matrix. This one performs a projection onto the axis at a 30° angle above the horizontal axis:" ] @@ -3793,14 +4343,16 @@ "cell_type": "code", "execution_count": 107, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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XXzxPefn8pN6uopj0psxcxGcHD77J228XUlQU3OZjauzpR5l5krme27lcXzy1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\nAAAAAElFTkSuQmCC\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3820,7 +4372,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "But this time, due to floating point rounding errors, NumPy manages to calculate an inverse (notice how large the elements are, though):" ] @@ -3829,7 +4384,9 @@ "cell_type": "code", "execution_count": 108, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3850,7 +4407,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "As you might expect, the dot product of a matrix by its inverse results in the identity matrix:\n", "\n", @@ -3863,7 +4423,9 @@ "cell_type": "code", "execution_count": 109, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3884,7 +4446,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Another way to express this is that the inverse of the inverse of a matrix $M$ is $M$ itself:\n", "\n", @@ -3895,7 +4460,9 @@ "cell_type": "code", "execution_count": 110, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3916,7 +4483,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Also, the inverse of scaling by a factor of $\\lambda$ is of course scaling by a factor or $\\frac{1}{\\lambda}$:\n", "\n", @@ -3931,14 +4501,16 @@ "cell_type": "code", "execution_count": 111, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3957,7 +4529,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Finally, a square matrix $H$ whose inverse is its own transpose is an **orthogonal matrix**:\n", "\n", @@ -3974,7 +4549,9 @@ "cell_type": "code", "execution_count": 112, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -3995,7 +4572,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Determinant\n", "The determinant of a square matrix $M$, noted $\\det(M)$ or $\\det M$ or $|M|$ is a value that can be calculated from its elements $(M_{i,j})$ using various equivalent methods. One of the simplest methods is this recursive approach:\n", @@ -4033,7 +4613,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "To get the determinant of a matrix, you can call NumPy's `det` function in the `numpy.linalg` module:" ] @@ -4042,7 +4625,9 @@ "cell_type": "code", "execution_count": 113, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4067,7 +4652,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "One of the main uses of the determinant is to *determine* whether a square matrix can be inversed or not: if the determinant is equal to 0, then the matrix *cannot* be inversed (it is a singular matrix), and if the determinant is not 0, then it *can* be inversed.\n", "\n", @@ -4078,7 +4666,9 @@ "cell_type": "code", "execution_count": 114, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4098,7 +4688,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "That's right, $F_{project}$ is singular, as we saw earlier." ] @@ -4107,7 +4700,9 @@ "cell_type": "code", "execution_count": 115, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4127,7 +4722,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "This determinant is suspiciously close to 0: it really should be 0, but it's not due to tiny floating point errors. The matrix is actually singular." ] @@ -4136,7 +4734,9 @@ "cell_type": "code", "execution_count": 116, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4156,14 +4756,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Perfect! This matrix *can* be inversed as we saw earlier. Wow, math really works!" ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "The determinant can also be used to measure how much a linear transformation affects surface areas: for example, the projection matrices $F_{project}$ and $F_{project\\_30}$ completely flatten the polygon $P$, until its area is zero. This is why the determinant of these matrices is 0. The shear mapping modified the shape of the polygon, but it did not affect its surface area, which is why the determinant is 1. You can try computing the determinant of a rotation matrix, and you should also find 1. What about a scaling matrix? Let's see:" ] @@ -4172,14 +4778,16 @@ "cell_type": "code", "execution_count": 117, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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EeXkQ5eVFCAZrEQwGUVRUZNu1UDNh8mNzPtjMTXH6tG7YVvZEc2KfUVdXBEC1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\ngg==\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -4198,7 +4806,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "We rescaled the polygon by a factor of 1/2 on both vertical and horizontal axes so the surface area of the resulting polygon is 1/4$^{th}$ of the original polygon. Let's compute the determinant and check that:" ] @@ -4207,7 +4818,9 @@ "cell_type": "code", "execution_count": 118, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4227,7 +4840,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Correct!\n", "\n", @@ -4238,7 +4854,9 @@ "cell_type": "code", "execution_count": 119, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4258,7 +4876,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Composing linear transformations\n", "Several linear transformations can be chained simply by performing multiple dot products in a row. For example, to perform a squeeze mapping followed by a shear mapping, just write:" @@ -4268,7 +4889,9 @@ "cell_type": "code", "execution_count": 120, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [], "source": [ @@ -4277,7 +4900,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Since the dot product is associative, the following code is equivalent:" ] @@ -4286,7 +4912,9 @@ "cell_type": "code", "execution_count": 121, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [], "source": [ @@ -4295,7 +4923,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Note that the order of the transformations is the reverse of the dot product order.\n", "\n", @@ -4306,7 +4937,9 @@ "cell_type": "code", "execution_count": 122, "metadata": { - "collapsed": true + "collapsed": true, + "deletable": true, + "editable": true }, "outputs": [], "source": [ @@ -4316,14 +4949,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "From now on we can perform both transformations in just one dot product, which can lead to a very significant performance boost." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "What if you want to perform the inverse of this double transformation? Well, if you squeezed and then you sheared, and you want to undo what you have done, it should be obvious that you should unshear first and then unsqueeze. In more mathematical terms, given two invertible (aka nonsingular) matrices $Q$ and $R$:\n", "\n", @@ -4336,7 +4975,9 @@ "cell_type": "code", "execution_count": 123, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4357,7 +4998,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Singular Value Decomposition\n", "It turns out that any $m \\times n$ matrix $M$ can be decomposed into the dot product of three simple matrices:\n", @@ -4374,7 +5018,9 @@ "cell_type": "code", "execution_count": 124, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4398,7 +5044,9 @@ "cell_type": "code", "execution_count": 125, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4418,7 +5066,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Note that this is just a 1D array containing the diagonal values of Σ. To get the actual matrix Σ, we can use NumPy's `diag` function:" ] @@ -4427,7 +5078,9 @@ "cell_type": "code", "execution_count": 126, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4449,7 +5102,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Now let's check that $U \\cdot \\Sigma \\cdot V^T$ is indeed equal to `F_shear`:" ] @@ -4458,7 +5114,9 @@ "cell_type": "code", "execution_count": 127, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4481,7 +5139,9 @@ "cell_type": "code", "execution_count": 128, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4502,7 +5162,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "It worked like a charm. Let's apply these transformations one by one (in reverse order) on the unit square to understand what's going on. First, let's apply the first rotation $V^T$:" ] @@ -4511,14 +5174,16 @@ "cell_type": "code", "execution_count": 129, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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WLH8XcVlZ1BEVh/L9IpFQOiiuSmXMAeX7RQpWSDpIZwJxNWkSrFrlvxn3xAed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\nIiIl7P8DykDSSJg1BVAAAAAASUVORK5CYII=\n", "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -4533,7 +5198,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Now let's rescale along the vertical and horizontal axes using $\\Sigma$:" ] @@ -4542,14 +5210,16 @@ "cell_type": "code", "execution_count": 130, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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peW0UZFvHq6ElnurTSSye0ejbxp5sSi+0OvwH/ncaHV0xg27n/Rc0cu3Fc3ng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\nUiqK/R9kJsBsOB4OigAAAABJRU5ErkJggg==\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -4564,7 +5234,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Finally, we apply the second rotation $U$:" ] @@ -4573,14 +5246,16 @@ "cell_type": "code", "execution_count": 131, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { "data": { - "image/png": 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qVq58g4aGAey99xD69tUD4KPok5+EZ5/NzCOHhw/3NYG4xx/3PYXaPuPmkEP8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\nAElFTkSuQmCC\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -4595,14 +5270,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "And we can see that the result is indeed a shear mapping of the original unit square." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Eigenvectors and eigenvalues\n", "An **eigenvector** of a square matrix $M$ (also called a **characteristic vector**) is a non-zero vector that remains on the same line after transformation by the linear transformation associated with $M$. A more formal definition is any vector $v$ such that:\n", @@ -4624,7 +5305,9 @@ "cell_type": "code", "execution_count": 132, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4647,7 +5330,9 @@ "cell_type": "code", "execution_count": 133, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4668,7 +5353,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Indeed the horizontal vectors are stretched by a factor of 1.4, and the vertical vectors are shrunk by a factor of 1/1.4=0.714…, so far so good. Let's look at the shear mapping matrix $F_{shear}$:" ] @@ -4677,7 +5365,9 @@ "cell_type": "code", "execution_count": 134, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4700,7 +5390,9 @@ "cell_type": "code", "execution_count": 135, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4721,14 +5413,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "Wait, what!? We expected just one unit eigenvector, not two. The second vector is almost equal to $\\begin{pmatrix}-1 \\\\ 0 \\end{pmatrix}$, which is on the same line as the first vector $\\begin{pmatrix}1 \\\\ 0 \\end{pmatrix}$. This is due to floating point errors. We can safely ignore vectors that are (almost) colinear (ie. on the same line)." ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "## Trace\n", "The trace of a square matrix $M$, noted $tr(M)$ is the sum of the values on its main diagonal. For example:" @@ -4738,7 +5436,9 @@ "cell_type": "code", "execution_count": 136, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4763,7 +5463,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "The trace does not have a simple geometric interpretation (in general), but it has a number of properties that make it useful in many areas:\n", "* $tr(A + B) = tr(A) + tr(B)$\n", @@ -4779,7 +5482,9 @@ "cell_type": "code", "execution_count": 137, "metadata": { - "collapsed": false + "collapsed": false, + "deletable": true, + "editable": true }, "outputs": [ { @@ -4799,30 +5504,42 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "deletable": true, + "editable": true + }, "source": [ "# What next?\n", - "This concludes this introduction to Linear Algeabra. Although these basics cover most of what you will need to know for Machine Learning, if you wish to go deeper into this topic there are many options available: Linear Algebra [books](http://linear.axler.net/), [Khan Academy](https://www.khanacademy.org/math/linear-algebra) lessons, or just [Wikipedia](https://en.wikipedia.org/wiki/Linear_algebra) pages. " + "This concludes this introduction to Linear Algebra. Although these basics cover most of what you will need to know for Machine Learning, if you wish to go deeper into this topic there are many options available: Linear Algebra [books](http://linear.axler.net/), [Khan Academy](https://www.khanacademy.org/math/linear-algebra) lessons, or just [Wikipedia](https://en.wikipedia.org/wiki/Linear_algebra) pages. " ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] } ], "metadata": { "kernelspec": { - "display_name": "Python 2", + "display_name": "Python 3", "language": "python", - "name": "python2" + "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 2 + "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython2", - "version": "2.7.11" + "pygments_lexer": "ipython3", + "version": "3.5.3" }, "toc": { "toc_cell": false,