{
"cells": [
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["%%html\n<link href=\"https:\/\/pretextbook.org\/beta\/mathbook-content.css\" rel=\"stylesheet\" type=\"text\/css\" \/>\n<link href=\"https:\/\/aimath.org\/mathbook\/mathbook-add-on.css\" rel=\"stylesheet\" type=\"text\/css\" \/>\n<style>.subtitle {font-size:medium; display:block}<\/style>\n<link href=\"https:\/\/fonts.googleapis.com\/css?family=Open+Sans:400,400italic,600,600italic\" rel=\"stylesheet\" type=\"text\/css\" \/>\n<link href=\"https:\/\/fonts.googleapis.com\/css?family=Inconsolata:400,700&subset=latin,latin-ext\" rel=\"stylesheet\" type=\"text\/css\" \/><!-- Hide this cell. -->\n<script>\nvar cell = $(\".container .cell\").eq(0), ia = cell.find(\".input_area\")\nif (cell.find(\".toggle-button\").length == 0) {\nia.after(\n    $('<button class=\"toggle-button\">Toggle hidden code<\/button>').click(\n        function (){ ia.toggle() }\n        )\n    )\nia.hide()\n}\n<\/script>\n"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["**Important:** to view this notebook properly you will need to execute the cell above, which assumes you have an Internet connection.  It should already be selected, or place your cursor anywhere above to select.  Then press the \"Run\" button in the menu bar above (the right-pointing arrowhead), or press Shift-Enter on your keyboard."]},
{"cell_type":"markdown", "metadata":{}, "source":["$\n\\newcommand{\\lt}{<}\n\\newcommand{\\gt}{>}\n\\newcommand{\\amp}{&}\n$"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><h2 class=\"heading\"><span class=\"title\">Sage and Linear Algebra Worksheet:<\/span> <span class=\"subtitle\">FCLA Section LT<\/span><\/h2><div class=\"author\"><div class=\"author-name\">Robert Beezer<\/div><div class=\"author-info\">Department of Mathematics and Computer Science<br \/>University of Puget Sound<\/div><\/div><div class=\"date\">Fall 2019<\/div><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-1\">Sage has very capable linear transformations from $\\mathbb{Q}^n$ to $\\mathbb{Q}^m\\text{.}$<\/p><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><h6 class=\"heading hide-type\"><span class=\"type\">Section<\/span> <span class=\"codenumber\">1<\/span> <span class=\"title\">Creation via Symbolic Functions<\/span><\/h6><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-2\">Start with a symbolic function.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["var('x1 x2 x3 x4')\nf(x1, x2, x3, x4) = (x1 + 2*x2 + x3 + 5*x4, x1 + 5*x2 + 4*x3 + 8*x4, -x2 - x3 - x4)"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-3\">Then specify the domain and codomain.  We need to be careful about how <code class=\"code-inline tex2jax_ignore\">T<\/code> prints, Sage likes rows.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["T = linear_transformation(QQ^4, QQ^3, f)\nT"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-4\">At a most basic level, <code class=\"code-inline tex2jax_ignore\">T<\/code> behaves as a function.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["u = random_vector(ZZ, 4, x=-9, y=9).change_ring(QQ)\nu, T(u)"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-5\">We can check Theorem LTTZZ, zero goes to zero.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["z4 = zero_vector(QQ, 4)\nz3 = zero_vector(QQ, 3)\nz4, T(z4), T(z4) == z3"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><h6 class=\"heading hide-type\"><span class=\"type\">Section<\/span> <span class=\"codenumber\">2<\/span> <span class=\"title\">Creation via Matrices<\/span><\/h6><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-6\">We can also create a linear transformation from a matrix, as in Theorem MBLT, with one caveat.  For a matrix $A\\text{,}$ the default is to create the function $T(\\mathbf{v})=\\mathbf{v}A\\text{.}$  The keywords option <code class=\"code-inline tex2jax_ignore\">side='right'<\/code> will indicate that we want to put the vector on the right of the matrix.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["A = matrix(QQ, [[1, 2, 1, 5], [1, 5, 4, 8], [0, -1, -1, -1]])\nS = linear_transformation(A, side='right')"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-7\">Notice that we do not have to specify the domain or codomain, these are inferred from the size and type of the matrix.  <code class=\"code-inline tex2jax_ignore\">S<\/code> is not new, it is exactly the linear transformation <code class=\"code-inline tex2jax_ignore\">T<\/code> above.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["S == T"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-8\">Again, notice how <code class=\"code-inline tex2jax_ignore\">S<\/code> prints — the matrix representation is the transpose of what we like to use.  This does not <em class=\"emphasis\">change<\/em> the linear transformation as a function, it just changes how we think about it (we like linear combinations of columns, Sage likes linear combinations of rows).<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["A, S"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><h6 class=\"heading hide-type\"><span class=\"type\">Section<\/span> <span class=\"codenumber\">3<\/span> <span class=\"title\">Creation via Values on a Basis<\/span><\/h6><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-9\">Starting with a domain and a codomain, we can provide a list of the images of basis vectors for the domain.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["v1 = vector(QQ, [1, 1, 0])\nv2 = vector(QQ, [2, 5, -1])\nv3 = vector(QQ, [1, 4, -1])\nv4 = vector(QQ, [5, 8, -1])\nR = linear_transformation(QQ^4, QQ^3, [v1, v2, v3, v4])"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-10\">That's right — same function again.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["R == T"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-11\">We can check how this construction works.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["d3 = R.domain().basis()[2]\nR(d3); R(d3) == v3"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-12\">We can give the domain an alternate basis and create a different linear transformation, despite seemingly having the same construction.  First we build the domain with a different user basis.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["u1 = vector(QQ, [1, 0, 0, 0])\nu2 = vector(QQ, [1, 1, 0, 0])\nu3 = vector(QQ, [1, 1, 1, 0])\nu4 = vector(QQ, [1, 1, 1, 1])\ndom4 = (QQ^4).subspace_with_basis([u1, u2, u3, u4])\ndom4"], "outputs":[]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["L = linear_transformation(dom4, QQ^3, [v1, v2, v3, v4])\nL"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-13\">Even though the matrix representation prints the same, this is not the same function, we will need ideas from Chapter R before we can understand the difference.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["R(u3), L(u3)"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-14\">This code should consistently return <code class=\"code-inline tex2jax_ignore\">False<\/code>.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["v = random_vector(QQ, 4)\nR(v) == L(v)"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><h6 class=\"heading hide-type\"><span class=\"type\">Section<\/span> <span class=\"codenumber\">4<\/span> <span class=\"title\">Basic Properties<\/span><\/h6><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-15\">Illustrations with <code class=\"code-inline tex2jax_ignore\">T<\/code>.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["T.domain()"], "outputs":[]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["T.codomain()"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-16\">A defining property, so always <code class=\"code-inline tex2jax_ignore\">True<\/code>.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["u = random_vector(QQ, 4)\nv = random_vector(QQ, 4)\nu, v, T(u+v) == T(u) + T(v)"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-17\">A defining property, so also always <code class=\"code-inline tex2jax_ignore\">True<\/code>.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["alpha = (QQ).random_element()\nu = random_vector(QQ, 4)\nalpha, u, T(alpha*u) == alpha*T(u)"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-18\">We can do “arithmetic” with linear transformations, though the addition seems to allow bad things to happen.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["R+S"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-19\">Scalar multiples also, and they are well-behaved.<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["12*T"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-20\">The following is wrong.  (In other words, there is a bug in Sage.)<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["P = R + L\nP"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-21\">As we can see…<\/p><\/div>"]},
{"cell_type":    "code", "execution_count":null, "metadata":{}, "source":["Q = L + R\nQ.is_equal_function(P)"], "outputs":[]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-22\">The problem is that Sage is simply adding the matrices representing the linear transformations, without checking that they are defined using domains with the <em class=\"emphasis\">same<\/em> basis.  We will understand the subtlety better in Chapter R.<\/p><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><\/div>"]},
{"cell_type":"markdown", "metadata":{}, "source":["<div class=\"mathbook-content\"><p id=\"p-23\">This work is Copyright 2016–2019 by Robert A. Beezer.  It is licensed under a <a class=\"external\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\" target=\"_blank\">Creative Commons Attribution-ShareAlike 4.0 International License<\/a>.<\/p><\/div>"]}
],
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