SLT.xml

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<!--   Sage and Linear Algebra Worksheets    -->
<!--            Robert A. Beezer             -->
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    <article xml:id="SLT">
        <title>Sage and Linear Algebra Worksheet</title>
        <subtitle>FCLA Section SLT</subtitle>

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        <frontmatter>
            <xi:include href="../header.xml" />
        </frontmatter>

        <section>
            <title>Surjective Linear Transformations</title>

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

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

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

            <p>The range is known in Sage as the <q>image.</q>  For a surjective linear transformation, it will be the entire codomain.  Note that the image is a vector space.</p>

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

            <sage><input>
            T.image() == T.codomain()
            </input></sage>

            <sage><input>
            B = matrix(QQ, [[1, -2, 0, 3], [3, -5, 1, 7], [-1, 4, 2, -7]])
            S = linear_transformation(QQ^4, QQ^3, B, side='right')
            </input></sage>

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

            <sage><input>
            IM = S.image()
            IM
            </input></sage>

            <sage><input>
            IM == S.codomain()
            </input></sage>
        </section>

        <section>
            <title>Pre-Images</title>

            <exercise>
                <statement>
                    <p>We can create inputs associated with any output.  First, we make an arbitrary output, but make sure it really is an output, as a linear combination of a basis of the image (see basis above).  We print the two vectors in the opposite of what we would consider the <q>normal</q> order.</p>
                </statement>
            </exercise>

            <sage><input>
            bas = IM.basis()
            out = ()*bas[0] + ()*bas[1]
            inp = S.preimage_representative(out)
            out, inp
            </input></sage>

            <p>A check on our work.</p>

            <sage><input>
            S(inp)
            </input></sage>

            <exercise>
                <statement>
                <p>We can make other inputs, using the kernel.</p>

                <p>Any value of <c>new_inp</c> is in the preimage of <c>out</c>, and every element of the preimage can be built this way.  Notice the role the kernel plays, much like in the worksheet about injective linear transformations.</p>
                </statement>
            </exercise>

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

            <sage><input>
            z = K.random_element()
            new_inp = inp + z
            new_inp, S(new_inp)
            </input></sage>

            <exercise>
                <statement>
                <p>Elements outside the range (image) will have empty preimages.  We mildly <q>wreck</q> an element of the range.</p>

                <p>With two initial entries determined by the zeros and ones in the basis vectors, the third entry must be determined, so we can <q>twiddle</q> it just a bit to obtain a vector of the codomain that lies outside the range.  We will ask Sage for a pre-image representative anyway and see what happens.</p>
                </statement>
            </exercise>

            <sage><input>
            in_range = ()*bas[0] + ()*bas[1]
            in_range
            </input></sage>

            <sage><input>
            outside_range = vector(QQ, [ , , ])
            S.preimage_representative(outside_range)
            </input></sage>
        </section>

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    </article>
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