MR.xml

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

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

        <section>
            <title>A Linear Transformation and some Bases</title>

            <p>In this section we define a linear transformation from <m>\mathbb{C}^6</m> to <m>\mathbb{C}^4</m>.  The definition is a <m>4\times 6</m> matrix of rank <m>4</m> that we will use to multiply input vectors with a matrix-vector product.  It is not important if the linear transformation is injective and/or surjective.</p>

            <sage><input>
            m, n = 4, 6
            A = random_matrix(QQ, m, n, algorithm='echelonizable', rank=m, upper_bound=9)
            A
            </input></sage>

            <sage><input>
            T = linear_transformation(A, side='right')
            T
            </input></sage>

            <p>And we construct two random bases, one for the domain and one for the codomain, extracted from the columns of unimodular matrices.</p>

            <sage><input>
            Dmat = random_matrix(QQ, n, n, algorithm='unimodular', upper_bound=9)
            D = Dmat.columns()
            D
            </input></sage>


            <sage><input>
            Cmat = random_matrix(QQ, m, m, algorithm='unimodular', upper_bound=9)
            C = Cmat.columns()
            C
            </input></sage>
        </section>

        <section>
            <title>A Matrix Representation, Old Style</title>

            <p>We will coordinatize the outputs of the linear transformation, for inputs from the basis of the domain, relative to the basis of the codomain, and pack them in a matrix.</p>

            <p>Outputs on the domain basis first.</p>

            <sage><input>
            outputs = [T(b) for b in D]
            outputs
            </input></sage>

            <p>We make a vector space for the codomain, with the desired basis.</p>

            <sage><input>
            VC = (QQ^m).subspace_with_basis(C)
            VC
            </input></sage>

            <p>Now, coordinate versions of the outputs.</p>

            <sage><input>
            coord = [VC.coordinate_vector(t) for t in outputs]
            coord
            </input></sage>

            <p>And then we pack them into a matrix.</p>

            <sage><input>
            rep = column_matrix(coord)
            rep
            </input></sage>

            <p>Does the representation <q>work</q> as it should?  We need a vector space for the domain before we can check.</p>

            <sage><input>
            VD = (QQ^n).subspace_with_basis(D)
            VD
            </input></sage>

            <p>OK, coordinatize input, multiply by representation matrix, un-coordinatize (linear combination).  This is the fundamental Theorem FTMR at work.</p>

            <sage><input>
            u = random_vector(QQ, 6)
            out = VC.linear_combination_of_basis(rep*VD.coordinate_vector(u))
            u, out, T(u) ==  out
            </input></sage>

            <p>Nice.</p>
        </section>

        <section>
            <title>A Matrix Representation, Sage Style</title>

            <p>Now we do it Sage style.  The matrix of the linear transformation first, as we like to see it.</p>

            <sage><input>
            T.matrix(side='right')
            </input></sage>

            <p>Now we replace the domain and codomain with new vector spaces, carrying different bases.  We are not really <q>restricting</q> the domain and codomain, we are replacing them by the same vector space, but each has a different basis.</p>

            <sage><input>
            S = T.restrict_domain(VD).restrict_codomain(VC)
            rep2 = S.matrix(side='right')
            rep2
            </input></sage>

            <p>Bingo!  This is representation we found above.  A one-liner in Sage.</p>

            <sage><input>
            rep2 == rep
            </input></sage>
        </section>

        <section>
            <title>Sneak Preview</title>

            <p>Ponder the following computation <mdash /> matrix representations and nonsingular matrices with columns from the two bases.</p>

            <sage><input>
            S.matrix(side='right') - Cmat.inverse()*T.matrix(side='right')*Dmat
            </input></sage>

            <p>Notice how Sage is conflicted about if <c>S</c> and <c>T</c> are equal or not.</p>

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

            <sage><input>
            S.is_equal_function(T)
            </input></sage>
        </section>

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    </article>
</pretext>