ILT.xml

<?xml version="1.0" encoding="UTF-8" ?>

<!--   Sage and Linear Algebra Worksheets    -->
<!--            Robert A. Beezer             -->
<!--  Copyright 2017-2019 License: CC BY-SA  -->
<!--  See COPYING for more information       -->

<pretext  xmlns:xi="http://www.w3.org/2001/XInclude">

    <xi:include href="../worksheetinfo.xml" />

    <article xml:id="ILT">
        <title>Sage and Linear Algebra Worksheet</title>
        <subtitle>FCLA Section ILT</subtitle>

        <!-- header inclusion needs -xinclude switch on xsltproc -->
        <frontmatter>
            <xi:include href="../header.xml" />
        </frontmatter>

        <section>
            <title>Injective Linear Transformations</title>

            <p>Two carefully-crafted linear transformations: <c>T</c> is injective, <c>S</c> is not.</p>

            <sage><input>
            A = matrix(QQ, [[1, 2, 2], [3, 7, 6], [1, 2, 1], [2, 5, 7]])
            T = linear_transformation(QQ^3, QQ^4, A, side='right')
            </input></sage>

            <sage><input>
            T.is_injective()
            </input></sage>

            <sage><input>
            T.kernel()
            </input></sage>

            <sage><input>
            B = matrix(QQ, [[0, 1, -2], [-1, 1, 3], [-2, 5, 0], [0, 2, -4]])
            S = linear_transformation(QQ^3, QQ^4, B, side='right')
            </input></sage>

            <sage><input>
            S.is_injective()
            </input></sage>

            <sage><input>
            K = S.kernel()
            K
            </input></sage>

            <p>We create two different inputs, which differ by a random vector from the kernel (which we hope is not simply the zero vector, a distinct possibility).  We always get the same output from <c>S</c>, predictably.  If we try this with <c>T</c> then the kernel vector is <em>always</em> the zero vector and the demonstration is very uninteresting.</p>

            <sage><input>
            z = K.random_element()
            u = random_vector(QQ, 3)
            w = u + z
            u, w, S(u), S(w), S(u) == S(w)
            </input></sage>

        </section>

        <xi:include href="../legal.xml" />

    </article>
</pretext>