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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
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| "# Assignment 3 - Linear Transformations\n", | |
| "\n", | |
| "Please check the assignment here:\n", | |
| "{URL}\n", | |
| "\n", | |
| "---\n", | |
| "\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "### Sessions 8.2 & 9.1: Fundamental Subspace and Graphs\n", | |
| "\n", | |
| "__1.__ Find the dimension and a basis for each of the four fundamental subspaces of the following matrices." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "#### Matrix A" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| ], | |
| "source": [ | |
| "A = matrix([[1, 2, 0, 1], [0, 1, 1, 0], [1, 2, 0, 1]])" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", | |
| "1 & 0 & -2 & 1 \\\\\n", | |
| "0 & 1 & 1 & 0 \\\\\n", | |
| "0 & 0 & 0 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[ 1 0 -2 1]\n", | |
| "[ 0 1 1 0]\n", | |
| "[ 0 0 0 0]" | |
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| "execution_count": 2, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | |
| "1 & 0 & 1 \\\\\n", | |
| "0 & 1 & 0 \\\\\n", | |
| "0 & 0 & 0 \\\\\n", | |
| "0 & 0 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[1 0 1]\n", | |
| "[0 1 0]\n", | |
| "[0 0 0]\n", | |
| "[0 0 0]" | |
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| }, | |
| "execution_count": 2, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}2</script></html>" | |
| ], | |
| "text/plain": [ | |
| "2" | |
| ] | |
| }, | |
| "execution_count": 2, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "show(A.rref()); show(A.transpose().rref()); show(rank(A))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "We retrieve the bases of _Col(A)_ and _Row(A)_ by the pivot columns and rows of the RREF, above.\n", | |
| "\n", | |
| "$Col(A) = \\{(1, 0, 1), (2, 1, 2)\\}.$\n", | |
| "\n", | |
| "$Row(A) = \\{(1, 2, 0, 1), (0, 1, 1, 0)\\}.$\n", | |
| "\n", | |
| "$Null(A) = \\{(2, -1, 1, 0), (-1, 0, 0, 1)\\} \\iff A\\vec{x}=0 \\iff \\begin{bmatrix} x\\\\ y \\\\ t \\\\ s \\end{bmatrix}= \\begin{bmatrix} 2t - s\\\\ -t \\\\ t \\\\ s \\end{bmatrix} = t\\begin{bmatrix} 2\\\\ -1 \\\\ 1 \\\\ 0 \\end{bmatrix} + s \\begin{bmatrix} -1\\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}$.\n", | |
| "\n", | |
| "$LeftNull(A) = \\{(-1, 0, 1)\\} \\iff A^T\\vec{y}=0 \\iff \\begin{bmatrix} x\\\\ y \\\\ t \\end{bmatrix} = \\begin{bmatrix} -t \\\\ 0 \\\\ t \\end{bmatrix} = t\\begin{bmatrix} -1\\\\ 0 \\\\ 1 \\end{bmatrix}$.\n", | |
| "\n", | |
| "We can verify the results via dimensionality counting.\n", | |
| "\n", | |
| "$A \\in \\mathbb{M}_{3\\times4}$. \n", | |
| "\n", | |
| "$dim(Col(A)) + dim(LeftNull(A)) = 2 + 1 = 3 = m.$ \n", | |
| "\n", | |
| "$dim(Row(A)) + dim(Null(A)) = 2 + 2 = 4 = n.$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "### Matrix U" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| ], | |
| "source": [ | |
| "U = matrix([[1, 2, 0, 1], [0, 1, 1, 0], [0, 0, 0, 0]])" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 4, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", | |
| "1 & 0 & -2 & 1 \\\\\n", | |
| "0 & 1 & 1 & 0 \\\\\n", | |
| "0 & 0 & 0 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[ 1 0 -2 1]\n", | |
| "[ 0 1 1 0]\n", | |
| "[ 0 0 0 0]" | |
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| "execution_count": 4, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | |
| "1 & 0 & 0 \\\\\n", | |
| "0 & 1 & 0 \\\\\n", | |
| "0 & 0 & 0 \\\\\n", | |
| "0 & 0 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
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| "text/plain": [ | |
| "[1 0 0]\n", | |
| "[0 1 0]\n", | |
| "[0 0 0]\n", | |
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| "execution_count": 4, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}2</script></html>" | |
| ], | |
| "text/plain": [ | |
| "2" | |
| ] | |
| }, | |
| "execution_count": 4, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "show(U.rref()); show(U.transpose().rref()); show(rank(U))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "Likewise,\n", | |
| "\n", | |
| "$Col(A) = \\{(1, 0, 0), (2, 1, 0)\\}.$\n", | |
| "\n", | |
| "$Row(A) = \\{(1, 2, 0, 1), (0, 1, 1, 0)\\}.$\n", | |
| "\n", | |
| "$Null(A) = \\{(2, -1, 1, 0), (-1, 0, 0, 1)\\} \\iff A\\vec{x}=0 \\iff \\begin{bmatrix} x\\\\ y \\\\ t \\\\ s \\end{bmatrix}= \\begin{bmatrix} 2t - s\\\\ -t \\\\ t \\\\ s \\end{bmatrix} = t\\begin{bmatrix} 2\\\\ -1 \\\\ 1 \\\\ 0 \\end{bmatrix} + s \\begin{bmatrix} -1\\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}$.\n", | |
| "\n", | |
| "$LeftNull(A) = \\{(0, 0, 1)\\} \\iff A^T\\vec{y}=0 \\iff \\begin{bmatrix} x\\\\ y \\\\ t \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\\\ t \\end{bmatrix} = t\\begin{bmatrix} 0\\\\ 0 \\\\ 1 \\end{bmatrix}$.\n", | |
| "\n", | |
| "We can verify the results via dimensionality counting.\n", | |
| "\n", | |
| "$A \\in \\mathbb{M}_{3\\times4}$. \n", | |
| "\n", | |
| "$dim(Col(A)) + dim(LeftNull(A)) = 2 + 1 = 3 = m.$ \n", | |
| "\n", | |
| "$dim(Row(A)) + dim(Null(A)) = 2 + 2 = 4 = n.$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "_Compare the results. How are these matrices related? How are their subspaces related?_\n", | |
| "\n", | |
| "The matrices are row equivalent. As a result, their row and null spaces are the same. Their column and left-null spaces are in turn different, although they preserve the same dimensisons." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "__2. Loops and Trees__ Draw a graph with numbered nodes and numbered, directed edges whose incidence matrix is" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 5, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", | |
| "-1 & 1 & 0 & 0 \\\\\n", | |
| "-1 & 0 & 1 & 0 \\\\\n", | |
| "0 & 1 & 0 & -1 \\\\\n", | |
| "0 & 0 & -1 & 1\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[-1 1 0 0]\n", | |
| "[-1 0 1 0]\n", | |
| "[ 0 1 0 -1]\n", | |
| "[ 0 0 -1 1]" | |
| ] | |
| }, | |
| "execution_count": 5, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", | |
| "1 & 0 & 0 & -1 \\\\\n", | |
| "0 & 1 & 0 & -1 \\\\\n", | |
| "0 & 0 & 1 & -1 \\\\\n", | |
| "0 & 0 & 0 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[ 1 0 0 -1]\n", | |
| "[ 0 1 0 -1]\n", | |
| "[ 0 0 1 -1]\n", | |
| "[ 0 0 0 0]" | |
| ] | |
| }, | |
| "execution_count": 5, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "A = matrix([[-1, 1, 0, 0], [-1, 0, 1, 0], [0, 1, 0, -1], [0, 0, -1, 1]])\n", | |
| "show(A); show(A.rref())" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| ], | |
| "source": [ | |
| "import networkx as nx\n", | |
| "import matplotlib.pyplot as plt\n", | |
| "import warnings\n", | |
| "\n", | |
| "\n", | |
| "warnings.filterwarnings(\"ignore\")" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": 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", | |
| "text/plain": [ | |
| "<Figure size 432x288 with 1 Axes>" | |
| ] | |
| }, | |
| "execution_count": 2, | |
| "metadata": { | |
| "image/png": { | |
| "height": 248, | |
| "width": 380 | |
| }, | |
| "needs_background": "light" | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "edges = [(1, 2), (1, 3), (4, 2), (3, 4)]\n", | |
| "G = nx.DiGraph()\n", | |
| "G.add_edges_from(edges)\n", | |
| "pos = nx.spring_layout(G)\n", | |
| "fig, ax = plt.subplots()\n", | |
| "nx.draw_networkx_nodes(G, pos, node_size = 500, node_color=\"orange\")\n", | |
| "nx.draw_networkx_labels(G, pos)\n", | |
| "nx.draw_networkx_edges(G, pos, arrows=True, width=1.5)\n", | |
| "ax.set_axis_off()\n", | |
| "plt.show()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "_(a) Describe the loops in your graph._\n", | |
| "\n", | |
| "We can note the existence of a loop (ignoring the direction of the edges) present in graph _A_, since we can depart from any node and arrive back at the same node without passing through any edge more than once. \n", | |
| "\n", | |
| "_(b)_ Put A in RREF by hand. Draw the associated graph at each step. The final graph should be a spanning tree (i.e., a graph without loops that includes each node).\n", | |
| "\n", | |
| "$$\n", | |
| "\\begin{bmatrix}\n", | |
| "-1 & 1 & 0 & 0 \\\\ \n", | |
| "-1 & 0 & 1 & 0 \\\\ \n", | |
| "0 & 1 & 0 & -1 \\\\\n", | |
| "0 & 0 & -1 & 1\n", | |
| "\\end{bmatrix}\\xrightarrow{R_2=R_2-R_1}\\begin{bmatrix}\n", | |
| "-1 & 1 & 0 & 0 \\\\ \n", | |
| "0 & -1 & 1 & 0 \\\\ \n", | |
| "0 & 1 & 0 & -1 \\\\\n", | |
| "0 & 0 & -1 & 1\n", | |
| "\\end{bmatrix}\\xrightarrow{R_3=R_3+R_2}\\begin{bmatrix}\n", | |
| "-1 & 1 & 0 & 0 \\\\ \n", | |
| "0 & -1 & 1 & 0 \\\\ \n", | |
| "0 & 0 & 1 & -1 \\\\\n", | |
| "0 & 0 & -1 & 1\n", | |
| "\\end{bmatrix}\\xrightarrow{R_4=R_4+R_3}\\begin{bmatrix}\n", | |
| "-1 & 1 & 0 & 0 \\\\ \n", | |
| "0 & -1 & 1 & 0 \\\\ \n", | |
| "0 & 0 & 1 & -1 \\\\\n", | |
| "0 & 0 & 0 & 0\n", | |
| "\\end{bmatrix}\\xrightarrow{R_2=R_3-R_2}\\begin{bmatrix}\n", | |
| "-1 & 1 & 0 & 0 \\\\ \n", | |
| "0 & 1 & 0 & -1 \\\\ \n", | |
| "0 & 0 & 1 & -1 \\\\\n", | |
| "0 & 0 & 0 & 0\n", | |
| "\\end{bmatrix}\\xrightarrow{R_1=R_2-R_1}\\begin{bmatrix}\n", | |
| "1 & 0 & 0 & -1 \\\\ \n", | |
| "0 & 1 & 0 & -1 \\\\ \n", | |
| "0 & 0 & 1 & -1 \\\\\n", | |
| "0 & 0 & 0 & 0\n", | |
| "\\end{bmatrix}$$\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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", | |
| "text/plain": [ | |
| "<Figure size 864x864 with 6 Axes>" | |
| ] | |
| }, | |
| "execution_count": 3, | |
| "metadata": { | |
| "image/png": { | |
| "height": 698, | |
| "width": 721 | |
| }, | |
| "needs_background": "light" | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "edges = {\n", | |
| " 1 : [(1, 2), (1, 3), (4, 2), (3, 4)],\n", | |
| " 2 : [(1, 2), (2, 3), (4, 2), (3, 4)],\n", | |
| " 3 : [(1, 2), (2, 3), (4, 3), (3, 4)],\n", | |
| " 4 : [(1, 2), (2, 3), (4, 3)],\n", | |
| " 5 : [(1, 2), (4, 2), (4, 3)],\n", | |
| " 6 : [(4, 1), (4, 2), (4, 3)]\n", | |
| "}\n", | |
| "\n", | |
| "graphs = [nx.DiGraph() for _ in range(len(edges))]\n", | |
| "for G, edge_list in zip(graphs, edges.values()):\n", | |
| " G.add_edges_from(edge_list)\n", | |
| "\n", | |
| "fig, axes = plt.subplots(3, 2, figsize=(12, 12))\n", | |
| "axes = [ax for ax_list in axes for ax in ax_list] # unlist\n", | |
| "\n", | |
| "def show(G, ax, title=\"\"): \n", | |
| " pos = nx.spring_layout(G)\n", | |
| " nx.draw_networkx_nodes(G, pos, node_size = 500, node_color=\"orange\", ax=ax)\n", | |
| " nx.draw_networkx_labels(G, pos, ax=ax)\n", | |
| " nx.draw_networkx_edges(G, pos, arrows=True, width=1.5, ax=ax)\n", | |
| " ax.set_axis_off()\n", | |
| " ax.set_title(title)\n", | |
| "\n", | |
| "for i, (G, ax) in enumerate(zip(graphs, axes), 1):\n", | |
| " show(G, ax, title=f\"A (step {i})\") \n", | |
| "\n", | |
| "plt.show()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "_(c) Conjecture the relationship between loops in the graph and the rows of the associated incidence matrix._\n", | |
| "\n", | |
| "Loops arise when rows in the incidence matrix are linearly dependent. When we reduce the incidence matrix to RREF, these loops are eliminated.\n", | |
| "\n", | |
| "_(d) Find a basis for Row(A). What is the relationship between Row(A) and the spanning tree you found?_\n", | |
| "\n", | |
| "Since the A RREF has three pivot colums, we can retrieve the basis of _Row(A)_ as the first three rows, i.e., $\\{[-1, 1, 0, 0], [-1, 0, 1, 0], [0, 1, 0, -1]\\}$. The incidence matrix of the spanning tree is the RREF form of the matrix composed via the basis of _Row(A)_. As a result, the spanning tree and basis have the same properties as row equivalent matrices. For example, they span the same vector space." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "### Sessions 9.2 & 10.1: Linear Transformations\n", | |
| "\n", | |
| "__1.__ Hip to be square.\n", | |
| "\n", | |
| "_(a) What is the shape of the transformed region when A = ...?_\n", | |
| "\n", | |
| "A parallelogram." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": 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", | |
| "text/plain": [ | |
| "Graphics object consisting of 6 graphics primitives" | |
| ] | |
| }, | |
| "execution_count": 1, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "A = matrix([[1, -2], [3, 4]])\n", | |
| "bottomleft = vector([0, 0])\n", | |
| "bottomright = vector([1, 0])\n", | |
| "topright = vector([1, 1])\n", | |
| "topleft = vector([0, 1])\n", | |
| "\n", | |
| "plot(arrow((0,0), A*bottomleft)) + \\\n", | |
| "plot(arrow((0,0), A*bottomright)) + \\\n", | |
| "plot(arrow((0,0), A*topright)) + \\\n", | |
| "plot(arrow((0,0), A*topleft)) + \\\n", | |
| "plot(arrow(A*topleft, A*topleft + A*bottomright)) + \\\n", | |
| "plot(arrow(A*bottomright, A*bottomright + A*topleft))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "_(b) In general, what is the shape of the transformed region?_\n", | |
| "\n", | |
| "A parallelogram.\n", | |
| "\n", | |
| "_(b) For which matrices A is that region a square?_\n", | |
| "\n", | |
| "For matrices that perform _only_ a rotation and/or an expansion/contraction of the vector space. In other words,\n", | |
| "\n", | |
| "$$A(k, \\theta) = \\begin{bmatrix} \n", | |
| "k\\cos\\theta & -k\\sin\\theta \\\\ k\\sin\\theta & k\\cos\\theta \n", | |
| "\\end{bmatrix}$$\n", | |
| "\n", | |
| "_(c) For which A is that region a line segment?_\n", | |
| "\n", | |
| "For $\\{A \\mid rank(A) = 1 \\}$. Matrices with linearly dependent row spaces will collapse the region into a line segment. If $\\text{rank}(A) = 0$, the region collapses into the origin. " | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "__2.__ All about the basis.\n", | |
| "\n", | |
| "_(a) Show that T is a linear transformation._\n", | |
| "\n", | |
| "We need only prove that T respects linearity under addition and scalar multiplication.\n", | |
| "\n", | |
| "Let $u = (u_1, u_2, u_3)$ and $v = (v_1, v_2, v_3)$. Then,\n", | |
| "\n", | |
| "$$\n", | |
| "\\begin{align}T(u + v) &= T(u_1 + v_1, u_2 + v_2, u_3 + v_3) \\\\ \n", | |
| "&= (u_1 + v_1 + 2u_2 + 2v_2 + 3u_3 + 3v_3, 3u_1 + 3v_1 − 2u_2 - 2v_2 + u_3 + v_3, 2u_1 + 2v_1 − 4u_2 - 4v_2 − 2u_3 -2v_3) \\\\\n", | |
| "&= (u_1 + 2u_2 + 3u_3 , 3u_1 − 2u_2 + u_3 , 2u_1 − 4u_2 − 2u_3 ) + (v_1+ 2v_2+ 3v_3, 3v_1- 2v_2+ v_3, 2v_1- 4v_2-2v_3) \\\\\n", | |
| "&= T(u) + T(v) \\\\\n", | |
| "\\end{align}$$\n", | |
| "\n", | |
| "Furthermore,\n", | |
| "\n", | |
| "$$\n", | |
| "\\begin{align} T(ku) &= T(ku_1, ku_2, ku_3) \\\\\n", | |
| "&= (ku_1 + 2ku_2 + 3ku_3 , 3ku_1 − 2ku_2 + ku_3 , 2ku_1 − 4ku_2 − 2ku_3 ) \\\\\n", | |
| "&= k(u_1 + 2u_2 + 3u_3 , 3u_1 − 2u_2 + u_3 , 2u_1 − 4u_2 − 2u_3 ) \\\\\n", | |
| "&= kT(u)\n", | |
| "\\end{align}$$\n", | |
| "\n", | |
| "Hence, T is a linear transformation." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 64, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | |
| "1 & 2 & 3 \\\\\n", | |
| "3 & -2 & 1 \\\\\n", | |
| "2 & -4 & -2\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[ 1 2 3]\n", | |
| "[ 3 -2 1]\n", | |
| "[ 2 -4 -2]" | |
| ] | |
| }, | |
| "execution_count": 64, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# (b) The combination describing each entry of the transformed vector forms the rows of A. Hence,\n", | |
| "A = matrix([[1, 2, 3], [3, -2, 1], [2, -4, -2]])\n", | |
| "show(A)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 65, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | |
| "1 & 0 & 1 \\\\\n", | |
| "0 & 1 & 1 \\\\\n", | |
| "0 & 0 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[1 0 1]\n", | |
| "[0 1 1]\n", | |
| "[0 0 0]" | |
| ] | |
| }, | |
| "execution_count": 65, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | |
| "1 & 0 & -1 \\\\\n", | |
| "0 & 1 & 1 \\\\\n", | |
| "0 & 0 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[ 1 0 -1]\n", | |
| "[ 0 1 1]\n", | |
| "[ 0 0 0]" | |
| ] | |
| }, | |
| "execution_count": 65, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# (c)\n", | |
| "show(A.rref())\n", | |
| "show(A.transpose().rref())" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "With the aid of the RREF, we find the bases for the fundamental subspaces.\n", | |
| "\n", | |
| "$Col(A) = \\{(1, 3, 2), (2, -2, -4)\\}.$\n", | |
| "\n", | |
| "$Row(A) = \\{(1, 2, 3), (3, -2, 1)\\}.$\n", | |
| "\n", | |
| "$Null(A) = \\{(-1, -1, 1)\\} \\iff A\\vec{x}=0 \\iff \\begin{bmatrix} x\\\\ y \\\\ t \\end{bmatrix}= \\begin{bmatrix} -t \\\\ -t \\\\ t \\end{bmatrix} = t\\begin{bmatrix} -1 \\\\ -1 \\\\ 1 \\end{bmatrix}$.\n", | |
| "\n", | |
| "$LeftNull(A) = \\{(1, -1, 1)\\} \\iff A^T\\vec{y}=0 \\iff \\begin{bmatrix} x\\\\ y \\\\ t \\end{bmatrix} = \\begin{bmatrix} t \\\\ -t \\\\ t \\end{bmatrix} = t\\begin{bmatrix} 1\\\\ -1 \\\\ 1 \\end{bmatrix}$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "_(d)_ \n", | |
| "\n", | |
| "i. Since _S_ contains three vectors, they span $\\mathbb{R}^3$ if they are linearly independent. Hence, if $\\det(A) \\neq 0$ or the rank of _A_ is three, S is a basis. " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 66, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}3</script></html>" | |
| ], | |
| "text/plain": [ | |
| "3" | |
| ] | |
| }, | |
| "execution_count": 66, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "S = matrix([[1, 0, -1], [0, 2, 3], [-1, 3, 0]]).transpose()\n", | |
| "assert det(S) != 0\n", | |
| "show(rank(S))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "Hence, S is a basis of $\\mathbb{R}^3$.\n", | |
| "\n", | |
| "ii. To find B, we have to apply T to the vectors in S and express the result in basis S." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 76, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | |
| "-\\frac{20}{11} & \\frac{142}{11} & \\frac{46}{11} \\\\\n", | |
| "\\frac{8}{11} & -\\frac{4}{11} & -\\frac{36}{11} \\\\\n", | |
| "\\frac{2}{11} & -\\frac{1}{11} & -\\frac{9}{11}\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[-20/11 142/11 46/11]\n", | |
| "[ 8/11 -4/11 -36/11]\n", | |
| "[ 2/11 -1/11 -9/11]" | |
| ] | |
| }, | |
| "execution_count": 76, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "T = lambda x: A*x\n", | |
| "\n", | |
| "# Last assignment, I created these functions to change vectors to other bases and\n", | |
| "# find the change-of-basis matrices. I'll reuse them here.\n", | |
| "\n", | |
| "def to_basis(v, basis):\n", | |
| " '''Receives the argument basis in matrix form.'''\n", | |
| " inv = basis.inverse()\n", | |
| " return list(inv*v)\n", | |
| "\n", | |
| "def find_P(basis_1, basis_2):\n", | |
| " return matrix([to_basis(v, basis_2) for v in basis_1.transpose()]).transpose()\n", | |
| "\n", | |
| "B = matrix([to_basis(T(u), S) for u in S.transpose()]).transpose()\n", | |
| "show(B)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "iii. Find the change-of-basis matrix P from S to E." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 77, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | |
| "1 & 0 & -1 \\\\\n", | |
| "0 & 2 & 3 \\\\\n", | |
| "-1 & 3 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[ 1 0 -1]\n", | |
| "[ 0 2 3]\n", | |
| "[-1 3 0]" | |
| ] | |
| }, | |
| "execution_count": 77, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "P = find_P(S, identity_matrix(3))\n", | |
| "show(P)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "iv. Show that $B = P_{−1}AP$. Explain the mechanics behind this relationship.\n", | |
| "\n", | |
| "See below. The relationship is grounded on the fact that a matrix of _T_ assumes a certain basis. _A_, for example, assumed the standard basis. If we want to transform a vector via the matrix, we want to guarantee that it is given in the appropriate basis. Hence, since _B_ is the transformation matrix in basis S, we can with no less of generality transform the input vector to the standard basis, apply A, and transform it back to basis S. Therefore,\n", | |
| "\n", | |
| "$$B = P_{S\\rightarrow E}AP_{E\\rightarrow S}= P^{-1}AP $$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 78, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | |
| "-\\frac{20}{11} & \\frac{142}{11} & \\frac{46}{11} \\\\\n", | |
| "\\frac{8}{11} & -\\frac{4}{11} & -\\frac{36}{11} \\\\\n", | |
| "\\frac{2}{11} & -\\frac{1}{11} & -\\frac{9}{11}\n", | |
| "\\end{array}\\right)</script></html>" | |
| ], | |
| "text/plain": [ | |
| "[-20/11 142/11 46/11]\n", | |
| "[ 8/11 -4/11 -36/11]\n", | |
| "[ 2/11 -1/11 -9/11]" | |
| ] | |
| }, | |
| "execution_count": 78, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "assert P.inverse() * A * P == B\n", | |
| "show(P.inverse() * A * P)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "### Sessions 10.2 & 11.1: The Fundamental Theorem of Linear Algebra\n", | |
| "\n", | |
| "__1.__\n", | |
| "\n", | |
| "(a): $\\mathcal{L} : P(2) \\rightarrow P(4)\\text{ given by }T(p(x)) = x^2p(x).$\n", | |
| "\n", | |
| "$p(x) = ax^2 + bx + c \\mid a, b, c \\in \\mathbb{R} \\iff p(x) \\in P(2).$ Let $p_1(x) = a_1x^2 + b_1x + c_1$ and $p_2(x) = a_2x^2 + b_2x + c_2$. Then,\n", | |
| "\n", | |
| "_Vector Addition:_ \n", | |
| "\n", | |
| "$$\\begin{align}\n", | |
| "T(p_1(x) + p_2(x)) &= T(x^2(a_1 + a_2) + x(b_1 + b_2) + (c_1 + c_2)) \\\\\n", | |
| "&= x^4(a_1 + a_2) + x^3(b_1 + b_2) + x^2(c_1 + c_2) \\\\\n", | |
| "&= a_1x^4 + b_1x^3 + c_1x^2 + a_2x^4 + b_2x^3 + c_2x^2 \\\\\n", | |
| "&= x^2(a_1x^2 + b_1x^1 + c_1) + \n", | |
| "\\end{align}$$\n", | |
| "\n", | |
| "_Scalar Multiplication:_\n", | |
| "\n", | |
| "$$\\begin{align}\n", | |
| "T(kp_1(x)) = T(a_1kx^2 + b_1kx + c_1k)= a_1kx^4 + b_1kx^3 + c_1kx^2 = kx^2(a_1x^2 + b_1x + c_1) = kT(p_1(x))\n", | |
| "\\end{align}$$\n", | |
| "\n", | |
| "Hence, $\\mathcal{L}$ is linear. Now, let us describe the coefficients of $P(2)$ and $P(4)$ as vectors in $\\mathbb{R}^3$ and $\\mathbb{R}^5$, respectively. Then, \n", | |
| "\n", | |
| "$$\\mathcal{L}(u) = Au \\mid A = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix}$$.\n", | |
| "\n", | |
| "_Im(T)_ = _Col(A)_ has thus a dimension of 3, and we can compose a basis from the pivot columns of _A_, such that $Im(T) = \\{(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0)\\},$ or the polynomials given by $ax^4 + bx^3 + cx^2 \\mid a, b, c \\in \\mathbb{R}$. Via dimensionality counting, we know that _dim(N(T))_ + _dim(Row(A))_ $= n = 3$. Since, _dim(Row(A))_ = _dim(Col(A))_ $= 3$, _dim(N(T))_ $= 0$. The basis is _N(T)_ is thus empty.\n", | |
| "\n", | |
| "(b): $\\mathcal{T} : \\mathcal{M}_{2\\times 2} \\rightarrow \\mathcal{M}_{2\\times 2} \\text{ given by }\\mathcal{T}(M) = M^T.$\n", | |
| "\n", | |
| "$m = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\mid a, b, c, d \\in \\mathbb{R} \\iff m \\in \\mathcal{M}_{2\\times 2}.$ Let $m_1 = \\begin{bmatrix} a_1 & b_1 \\\\ c_1 & d_1 \\end{bmatrix}$ and $m_2 = \\begin{bmatrix} a_2 & b_2 \\\\ c_2 & d_2 \\end{bmatrix}$. Then,\n", | |
| "\n", | |
| "_Vector Addition:_ \n", | |
| "\n", | |
| "$$\\begin{align}\n", | |
| "T(m_1 + m_2) &= T(\\begin{bmatrix} a_1 + a_2 & b_1 + b_2 \\\\ c_1 + c_2 & d_1 + d_2 \\end{bmatrix}) \\\\\n", | |
| "&= \\begin{bmatrix} a_1 + a_2 & c_1 + c_2 \\\\ b_1 + b_2 & d_1 + d_2 \\end{bmatrix} \\\\\n", | |
| "&= \\begin{bmatrix} a_1 & c_1 \\\\ b_1 & d_1 \\end{bmatrix} + \\begin{bmatrix} a_2 & c_2 \\\\ b_2 & d_2 \\end{bmatrix} \\\\\n", | |
| "&= T(m_1) + T(m_2)\n", | |
| "\\end{align}$$\n", | |
| "\n", | |
| "_Scalar Multiplication:_ \n", | |
| "\n", | |
| "$$\\begin{align}\n", | |
| "T(km_1) &= T(\\begin{bmatrix} ka_1 & kb_1 \\\\ kc_1& kd_1 \\end{bmatrix}) \\\\\n", | |
| "&= \\begin{bmatrix} ka_1 & kc_1 \\\\ kb_1 & kd_1 \\end{bmatrix} \\\\\n", | |
| "&= k\\begin{bmatrix} a_1 & c_1 \\\\ b_1 & d_1 \\end{bmatrix} \\\\\n", | |
| "&= kT(m_1)\n", | |
| "\\end{align}$$\n", | |
| "\n", | |
| "Hence, $\\mathcal{T}$ is linear. Now, let us describe the entries of $\\mathcal{M}_{2 \\times 2}$ as vectors in $\\mathbb{R}^4$. Then, \n", | |
| "\n", | |
| "$$\\mathcal{T}(m) = Am \\mid A = \\begin{bmatrix} 1 & 0 & 0 &0 \\\\ 0 &0& 1 & 0 \\\\ 0 & 1&0 & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix}$$.\n", | |
| "\n", | |
| "Similar to the above, since _Im(T)_ = _Col(A)_, the basis of _Col(A)_ is equal to the standard basis of $\\mathbb{R}^4$. _Im(T)_ thus spans $\\mathcal{M}_{2 \\times 2}$, and the basis may be given by $\\{\\begin{bmatrix} 1 & 0\\\\ 0 & 0 \\end{bmatrix}, \\begin{bmatrix} 0 & 1\\\\ 0 & 0 \\end{bmatrix}, \\begin{bmatrix} 0 & 0\\\\ 1 & 0 \\end{bmatrix}, \\begin{bmatrix} 0 & 0\\\\ 0 & 1 \\end{bmatrix}\\}$. Via dimensionality counting, we know that _dim(N(T))_ + _dim(Row(A))_ $= n = 4$. Since, _dim(Row(A))_ = _dim(Col(A))_ $= 4$, _dim(N(T))_ $= 0$. The basis is _N(T)_ is thus empty.\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "__2.__ Show that the following vector spaces are isomorphic.\n", | |
| "\n", | |
| "(a) Let $p(x) \\in P(n)$ be a polynomial of the form $p(x) = k_nx^n + k_{n-1}x^{n-1} + \\dots + k_0$. Furthermore, let $u \\in \\mathbb{R}^{n+1}$ be a vector of the form $u = (k_n, k_{n-1}, \\dots, k_0)$. Let $f : P(n) \\rightarrow \\mathbb{R}^{n+1} \\mid f(k_nx^n + k_{n-1}x^{n-1} + \\dots + k_0) = (k_n, k_{n-1}, \\dots, k_0)$ and $g : \\mathbb{R}^{n+1} \\rightarrow P(n) \\mid f(k_n, k_{n-1}, \\dots, k_0) = k_nx^n + k_{n-1}x^{n-1} + \\dots + k_0.$ Since $g(f(p(x))) = p(x)$, $f$ is an invertible map. Hence, $P(n)$ and $\\mathbb{R}^{n+1}$ are isomorhpic." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "(b) Let $m \\in \\mathcal{M}_{n\\times m}$ be a matrix of the form $\\begin{bmatrix}k_{11} & \\dots & k_{1m} \\\\ \\vdots & \\ddots & \\\\ k_{n1} && k_{nm}\\end{bmatrix}$. Furthermore, let $u \\in \\mathbb{R}^{nm}$ be a vector of the form $u = (k_1, \\dots, k_{nm})$. Let $f : \\mathcal{M}_{n\\times m} \\rightarrow \\mathbb{R}^{nm} \\mid f(\\begin{bmatrix}k_{11} & \\dots & k_{1m} \\\\ \\vdots & \\ddots & \\\\ k_{n1} && k_{nm}\\end{bmatrix}) = (k_1, \\dots, k_{nm})$ and $g : \\mathbb{R}^{nm} \\rightarrow \\mathcal{M}_{n\\times m} \\mid f(k_1, \\dots, k_{nm}) = \\begin{bmatrix}k_{11} & \\dots & k_{1m} \\\\ \\vdots & \\ddots & \\\\ k_{n1} && k_{nm}\\end{bmatrix}.$ Since $g(f(m)) = m$, $f$ is an invertible map. Hence, $\\mathcal{M}_{n\\times m}$ and $\\mathbb{R}^{nm}$ are isomorphic." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "(c) Let $p(x) \\in P(5)$ be a polynomial of the form $p(x) = k_5x^5 + k_4x^4 + \\dots + k_0$. Furthermore, let $m \\in \\mathcal{M}_{2\\times 3}$ be a matrix of the form $\\begin{bmatrix}k_0 & k_1 & k_2 \\\\ k_3 & k_4 & k_5\\end{bmatrix}$. Let $f : P(5) \\rightarrow \\mathcal{M}_{2\\times 3} \\mid f(k_5x^5 + k_4x^4 + \\dots + k_0) = \\begin{bmatrix}k_0 & k_1 & k_2 \\\\ k_3 & k_4 & k_5\\end{bmatrix}$ and $g : \\mathcal{M}_{2\\times 3} \\rightarrow P(5) \\mid f(\\begin{bmatrix}k_0 & k_1 & k_2 \\\\ k_3 & k_4 & k_5\\end{bmatrix}) = k_5x^5 + k_4x^4 + \\dots + k_0.$ Since $g(f(p(x))) = p(x)$, $f$ is an invertible map. Hence, $\\mathcal{M}_{2\\times 3}$ and $P(5)$ are isomorhpic." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "## Deep Dive" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "_1. Avoiding Carmageddon_\n", | |
| "\n", | |
| "(a) The potential at each node may be understood as the number of cars passing through each roundabout. The difference in potential on the edges could be interpreted as the number of cars going from one roundabout to the other.\n", | |
| "\n", | |
| "(b) We can number the nodes starting with $1$ at the top-left node and then proceed clockwise. The edge numbers can be inferred from the incidence matrix is given by \n", | |
| "\n", | |
| "$$A = \\begin{bmatrix} -1 & 1 & 0 & 0 & 0 & 0 \\\\ \n", | |
| "0 & -1 & 1 & 0 & 0 & 0 \\\\ \n", | |
| "0 & 0 & 1 & -1 & 0 & 0 \\\\ \n", | |
| "0 & 0 & 0 & -1 & 1 & 0 \\\\\n", | |
| "0 & 0 & 0 & 0 & -1 & 1 \\\\\n", | |
| "1 & 0 & 0 & 0 & 0 & -1 \\\\\n", | |
| "0 &-1 & 0 & 0 & 1 & 0\n", | |
| "\\end{bmatrix}$$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrr}\n", | |
| "-1 & 1 & 0 & 0 & 0 & 0 \\\\\n", | |
| "0 & -1 & 1 & 0 & 0 & 0 \\\\\n", | |
| "0 & 0 & 1 & -1 & 0 & 0 \\\\\n", | |
| "0 & 0 & 0 & -1 & 1 & 0 \\\\\n", | |
| "0 & 0 & 0 & 0 & -1 & 1 \\\\\n", | |
| "1 & 0 & 0 & 0 & 0 & -1 \\\\\n", | |
| "0 & -1 & 0 & 0 & 1 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
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| "text/html": [ | |
| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrr}\n", | |
| "1 & 0 & 0 & 0 & 0 & -1 \\\\\n", | |
| "0 & 1 & 0 & 0 & 0 & -1 \\\\\n", | |
| "0 & 0 & 1 & 0 & 0 & -1 \\\\\n", | |
| "0 & 0 & 0 & 1 & 0 & -1 \\\\\n", | |
| "0 & 0 & 0 & 0 & 1 & -1 \\\\\n", | |
| "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", | |
| "0 & 0 & 0 & 0 & 0 & 0\n", | |
| "\\end{array}\\right)</script></html>" | |
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| "[ 1 0 0 0 0 -1]\n", | |
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| "[ 0 0 0 1 0 -1]\n", | |
| "[ 0 0 0 0 1 -1]\n", | |
| "[ 0 0 0 0 0 0]\n", | |
| "[ 0 0 0 0 0 0]" | |
| ] | |
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| "execution_count": 2, | |
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| }, | |
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| "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrr}\n", | |
| "1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n", | |
| "0 & 1 & 0 & 0 & 0 & -1 & 1 \\\\\n", | |
| "0 & 0 & 1 & 0 & 0 & 1 & -1 \\\\\n", | |
| "0 & 0 & 0 & 1 & 0 & -1 & 1 \\\\\n", | |
| "0 & 0 & 0 & 0 & 1 & -1 & 0 \\\\\n", | |
| "0 & 0 & 0 & 0 & 0 & 0 & 0\n", | |
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| "[ 1 0 0 0 0 -1 0]\n", | |
| "[ 0 1 0 0 0 -1 1]\n", | |
| "[ 0 0 1 0 0 1 -1]\n", | |
| "[ 0 0 0 1 0 -1 1]\n", | |
| "[ 0 0 0 0 1 -1 0]\n", | |
| "[ 0 0 0 0 0 0 0]" | |
| ] | |
| }, | |
| "execution_count": 2, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "M = matrix([[-1, 1, 0, 0, 0, 0], [0, -1, 1, 0, 0, 0], [0, 0, 1, -1, 0, 0], [0, 0, 0, -1, 1, 0], [0, 0, 0, 0, -1, 1], [1, 0, 0, 0, 0, -1], [0, -1, 0, 0, 1, 0]])\n", | |
| "show(M)\n", | |
| "show(M.rref())\n", | |
| "show(M.transpose().rref())" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "The bases of the fundamental spaces are then, noting the pivot columns,\n", | |
| "\n", | |
| "$Col(A) = \\{(-1, 0, 0, 0, 0, 1, 0), (1, -1, 0, 0, 0, 0, -1), (0, 1, 1, 0, 0, 0, 0), (0, 0, -1, -1, 0, 0, 0), (0, 0, 0, 1, -1, 0, 1)\\}.$\n", | |
| "\n", | |
| "$Row(A) = \\{(-1, 1, 0, 0, 0, 0), (0, -1, 1, 0, 0, 0), (0, 0, 1, -1, 0, 0), (0, 0, 0, -1, 1, 0), (0, 0, 0, 0, -1, 1)\\}.$\n", | |
| "\n", | |
| "$Null(A) = \\{(1, 1, 1, 1, 1, 1)\\} \\iff A\\vec{x}=0 \\iff \\begin{bmatrix} x_1\\\\ x_2 \\\\ x_3 \\\\ x_4 \\\\ x_5 \\\\ t \\end{bmatrix}= \\begin{bmatrix} t\\\\ t \\\\ t \\\\ t \\\\ t \\\\ t \\end{bmatrix} = t\\begin{bmatrix} 1\\\\ 1 \\\\ 1 \\\\ 1 \\\\ 1 \\\\ 1 \\end{bmatrix}$.\n", | |
| "\n", | |
| "$LeftNull(A) = \\{(1, 1, -1, 1, 1, 1, 0), (0, -1, 1, -1, 0, 0, 1)\\} \\iff A^T\\vec{y}=0 \\iff \\begin{bmatrix} y_1 \\\\ y_2 \\\\ y_3 \\\\ y_4 \\\\ y_5 \\\\ t \\\\ s \\end{bmatrix} = \\begin{bmatrix} t\\\\ t - s \\\\ -t + s \\\\ t - s \\\\ t \\\\ t \\\\ s \\end{bmatrix} = t\\begin{bmatrix} 1\\\\ 1 \\\\ -1 \\\\ 1 \\\\ 1 \\\\ 1 \\\\ 0 \\end{bmatrix} + s\\begin{bmatrix} 0\\\\ - 1 \\\\ 1 \\\\ -1 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}$.\n", | |
| "\n", | |
| "We can verify the results via dimensionality counting.\n", | |
| "\n", | |
| "$A \\in \\mathbb{M}_{7\\times6}$. \n", | |
| "\n", | |
| "$dim(Col(A)) + dim(LeftNull(A)) = 5 + 2 = 7 = m.$ \n", | |
| "\n", | |
| "$dim(Row(A)) + dim(Null(A)) = 5 + 1 = 6 = n.$\n", | |
| "\n", | |
| "The column space represents the potential flows between the roundabouts, the nullspace are the number of cars in the roundabouts so that the differences between roundabouts are zero, the left-nullspace represents the potential flows into and out of the node being equal, and the row space describes the possible delta flows for each roundabout." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "(c) The key assumption is that the flows do not vary as a function of time, which is usually the case in real-life traffic. Furthermore, there is no stochastic component to the model, which is often characteristic of behavior in traffic as well.\n", | |
| "\n", | |
| "(d) Arguably, the important requirement of the traffic model is that the flow in and out of a roundabout be equal to zero, so that a traffic jam is not formed. The solution will exist in the row space of the incidence matrix where $A^Ty = d$, where _d_ accounts for the traffic flowing across the boundaries of the system. Thus,\n", | |
| "\n", | |
| "$$\n", | |
| "A^Ty = \n", | |
| "\\begin{bmatrix} -1 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n", | |
| "1 & -1 & 0 & 0 & 0 & 0 & -1 \\\\\n", | |
| "0 &1 & 1 & 0 & 0 & 0 & 0 \\\\\n", | |
| "0 & 0 & -1 & -1 & 0 & 0 & 0 \\\\\n", | |
| "0 & 0 & 0 & 1 & -1 & 0 & 1 \\\\\n", | |
| "0 & 0 & 0 & 0 & 1 & -1 & 0 \n", | |
| "\\end{bmatrix} \\begin{bmatrix}\n", | |
| "y_1 \\\\ y_2 \\\\ y_3 \\\\ y_4 \\\\ y_5 \\\\ y_6 \\\\ y_7 \n", | |
| "\\end{bmatrix} + \\begin{bmatrix}\n", | |
| "-200 \\\\ 300 \\\\ -400 \\\\ 400 \\\\ -350 \\\\ 250 \n", | |
| "\\end{bmatrix} = \\begin{bmatrix}\n", | |
| "0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \n", | |
| "\\end{bmatrix}\n", | |
| "$$\n", | |
| "|\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[[y1 == r2 - 200, y2 == -r1 + r2 + 100, y3 == r1 - r2 + 300, y4 == -r1 + r2 + 100, y5 == r2 - 250, y6 == r2, y7 == r1]]" | |
| ] | |
| }, | |
| "execution_count": 3, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "y1, y2, y3, y4, y5, y6, y7 = var('y1, y2, y3, y4, y5, y6, y7')\n", | |
| "u = vector([y1, y2, y3, y4, y5, y6, y7])\n", | |
| "flows = [200, -300, 400, -400, 350, -250]\n", | |
| "eqs = [lhand == rhand for lhand, rhand in zip(M.transpose() * u, flows)]\n", | |
| "solve(eqs, (y1, y2, y3, y4, y5, y6, y7))\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "We have two degrees of freedom. Assuming $r_1 = r_2 = 300$, we would have the following ordered sequence of traffic flows : (100, 100, 300, 100, 50, 300, 300). In this case, the traffic varies somewhat substantially from one road to the other, but otherwise the results are plausible and practical. We can increase traffic through the fifth road away from the third, sixth, and seventh.\n", | |
| "\n", | |
| "(e) The flows in roads six and seven are free variables, so presumably these could be disrupted without necessarily affecting the remainder of the traffic." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 4, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[]" | |
| ] | |
| }, | |
| "execution_count": 4, | |
| "metadata": { | |
| }, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "y1, y2, y3, y4, y5, y6, y7 = var('y1, y2, y3, y4, y5, y6, y7')\n", | |
| "u = vector([y1, y2, y3, y4, y5, y6, y7])\n", | |
| "flows = [200, -1000, 600, -400, 500, -100]\n", | |
| "eqs = [lhand == rhand for lhand, rhand in zip(M.transpose() * u, flows)]\n", | |
| "solve(eqs, (y1, y2, y3, y4, y5, y6, y7))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "(f) The system has no solution, indicating that there is no stable traffic within these roads that count take care of this flow. Hence, the proposed direction of the streets need to be optimized further." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "__2.__ To infinity, and beyond.\n", | |
| "\n", | |
| "(a) Let $p(x) \\in \\mathcal{P}(\\mathbb{R}) \\mid p(x) = \\sum_{i=0}^n a_ix^i, n \\in \\mathbb{R}$. Similarly for a polynomial $q(x) = \\sum_{i=0}^n b_ix^i$. Since $p(x) + q(x) = \\sum_{i=0}^n a_ix^i + \\sum_{i=0}^n b_ix^i = \\sum_{i=0}^n (a_i + b_i)x^i = (p+q)(x),$ $\\mathcal{P}(\\mathbb{R})$ is closed under vector addition. Similarly, $cp(x) = c\\sum_{i=0}^n a_ix^i = \\sum_{i=0}^n ca_ix^i = (cp)(x),$ showing that $\\mathcal{P}(\\mathbb{R})$ is closed under scalar multiplication.\n", | |
| "\n", | |
| "(b) Let us assume that there is a finite dimensional vector space $\\mathcal{V}$ such that $\\mathcal{V}$ and $\\mathcal{P}(\\mathbb{R})$ are isomorphic. If that is the case, then there is an invertible map $T$ such that $Im(T) = \\mathcal{P}(\\mathbb{R})$. However, if $\\mathcal{V}$ has dimension of $n$, it can span at most an n-dimensional vector space. Since $dim(Im(T)) = \\infty > n,$ the two vector spaces cannot be isomorphic.\n", | |
| "\n", | |
| "(c) $\\{k_ix^i : i \\in \\mathbb{N}, k_i \\neq 0 \\}$\n" | |
| ] | |
| } | |
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