LT.xml

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

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

        <introduction>
            <p>Sage has very capable linear transformations from <m>\mathbb{Q}^n</m> to <m>\mathbb{Q}^m</m>.</p>
        </introduction>

        <section>
            <title>Creation via Symbolic Functions</title>

            <p>Start with a symbolic function.</p>

            <!-- [[1, 2, 1, 5], [1, 5, 4, 8], [0, -1, -1, -1]] -->

            <sage><input>
            var('x1 x2 x3 x4')
            f(x1, x2, x3, x4) = (x1 + 2*x2 + x3 + 5*x4, x1 + 5*x2 + 4*x3 + 8*x4, -x2 - x3 - x4)
            </input></sage>

            <p>Then specify the domain and codomain.  We need to be careful about how <c>T</c> prints, Sage likes rows.</p>

            <sage><input>
            T = linear_transformation(QQ^4, QQ^3, f)
            T
            </input></sage>

            <p>At a most basic level, <c>T</c> behaves as a function.</p>

            <sage><input>
            u = random_vector(ZZ, 4, x=-9, y=9).change_ring(QQ)
            u, T(u)
            </input></sage>

            <p>We can check Theorem LTTZZ, zero goes to zero.</p>

            <sage><input>
            z4 = zero_vector(QQ, 4)
            z3 = zero_vector(QQ, 3)
            z4, T(z4), T(z4) == z3
            </input></sage>

        </section>

        <section>
            <title>Creation via Matrices</title>

            <p>We can also create a linear transformation from a matrix, as in Theorem MBLT, with one caveat.  For a matrix <m>A</m>, the default is to create the function <m>T(\mathbf{v})=\mathbf{v}A</m>.  The keywords option <c>side='right'</c> will indicate that we want to put the vector on the right of the matrix.</p>

            <sage><input>
            A = matrix(QQ, [[1, 2, 1, 5], [1, 5, 4, 8], [0, -1, -1, -1]])
            S = linear_transformation(A, side='right')
            </input></sage>

            <p>Notice that we do not have to specify the domain or codomain, these are inferred from the size and type of the matrix.  <c>S</c> is not new, it is exactly the linear transformation <c>T</c> above.</p>

            <sage><input>
            S == T
            </input></sage>

            <p>Again, notice how <c>S</c> prints <mdash /> the matrix representation is the transpose of what we like to use.  This does not <em>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>

            <sage><input>
            A, S
            </input></sage>

        </section>

        <section>
            <title>Creation via Values on a Basis</title>

            <p>Starting with a domain and a codomain, we can provide a list of the images of basis vectors for the domain.</p>

            <sage><input>
            v1 = vector(QQ, [1, 1, 0])
            v2 = vector(QQ, [2, 5, -1])
            v3 = vector(QQ, [1, 4, -1])
            v4 = vector(QQ, [5, 8, -1])
            R = linear_transformation(QQ^4, QQ^3, [v1, v2, v3, v4])
            </input></sage>

            <p>That's right <mdash /> same function again.</p>

            <sage><input>
            R == T
            </input></sage>

            <p>We can check how this construction works.</p>

            <sage><input>
            d3 = R.domain().basis()[2]
            R(d3); R(d3) == v3
            </input></sage>

            <p>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>

            <sage><input>
            u1 = vector(QQ, [1, 0, 0, 0])
            u2 = vector(QQ, [1, 1, 0, 0])
            u3 = vector(QQ, [1, 1, 1, 0])
            u4 = vector(QQ, [1, 1, 1, 1])
            dom4 = (QQ^4).subspace_with_basis([u1, u2, u3, u4])
            dom4
            </input></sage>

            <sage><input>
            L = linear_transformation(dom4, QQ^3, [v1, v2, v3, v4])
            L
            </input></sage>

            <p>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>

            <sage><input>
            R(u3), L(u3)
            </input></sage>

            <p>This code should consistently return <c>False</c>.</p>

            <sage><input>
            v = random_vector(QQ, 4)
            R(v) == L(v)
            </input></sage>

        </section>

        <section>
            <title>Basic Properties</title>

            <p>Illustrations with <c>T</c>.</p>

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


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

            <p>A defining property, so always <c>True</c>.</p>

            <sage><input>
            u = random_vector(QQ, 4)
            v = random_vector(QQ, 4)
            u, v, T(u+v) == T(u) + T(v)
            </input></sage>

            <p>A defining property, so also always <c>True</c>.</p>

            <sage><input>
            alpha = (QQ).random_element()
            u = random_vector(QQ, 4)
            alpha, u, T(alpha*u) == alpha*T(u)
            </input></sage>

            <p>We can do <q>arithmetic</q> with linear transformations, though the addition seems to allow bad things to happen.</p>

            <sage><input>
            R+S
            </input></sage>

            <p>Scalar multiples also, and they are well-behaved.</p>

            <sage><input>
            12*T
            </input></sage>

            <p>The following is wrong.  (In other words, there is a bug in Sage.)</p>

            <sage><input>
            P = R + L
            P
            </input></sage>

            <p>As we can see<ellipsis /></p>

            <sage><input>
            Q = L + R
            Q.is_equal_function(P)
            </input></sage>

            <p>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>same</em> basis.  We will understand the subtlety better in Chapter R.</p>

        </section>

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    </article>
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