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

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

        <section>
            <title>Vector Representations</title>

            <p>It is easy to form vector representations of vectors in <m>\mathbb{C}^n</m>.</p>

            <p>We get a nonstandard basis quickly from the columns of a nonsingular matrix.  The keyword <c>algorithm='unimodular'</c> requests a matrix with determinant <m>1</m>.</p>

            <sage><input>
            n = 6
            A = random_matrix(QQ, n, algorithm='unimodular', upper_bound=9)
            A
            </input></sage>

            <p>The columns of <c>A</c> become the <q>user basis</q> of a vector space.</p>

            <sage><input>
            B = A.columns()
            V = (QQ^n).subspace_with_basis(B)
            V
            </input></sage>


            <sage><input>
            u = random_vector(QQ, n)
            u
            </input></sage>

            <p>Now, we get values of the invertible linear transformation <m>\rho_B</m> with the Sage method <c>.coordinate_vector()</c> method of the vector space.</p>

            <sage><input>
            c = V.coordinate_vector(u)
            c
            </input></sage>

            <p>The inverse linear transformation is also available as the <c>.linear_combination_of_basis()</c> method of the vector space.</p>

            <sage><input>
            round_trip = V.linear_combination_of_basis(c)
            round_trip
            </input></sage>

            <p>And the automated check:</p>

            <sage><input>
            u == round_trip
            </input></sage>

            <p>Notice that this is something we could do <q>by hand</q> with just reduced row-echelon form. The coordinitization of <c>u</c> relative to the basis <c>B</c> is just a (unique) solution to a linear system.</p>

            <sage><input>
            aug = column_matrix(B + [u])
            aug.rref()
            </input></sage>

            <p>The following stanza will always return <c>True</c> as we <q>coordinatize</q> and then use the coordinates to form a linear combination of the basis.</p>

            <sage><input>
            w = random_vector(QQ, n)
            x = V.coordinate_vector(w)
            y = V.linear_combination_of_basis(x)
            y == w
            </input></sage>
        </section>

        <!-- Might be better to compute something of interest in abstract vector space by coordinatizing  -->
        <section>
            <title>Abstract Vector Spaces</title>

            <p>Sage does not implement abstract vector spaces.  It presumes we have <q>nice</q> standard bases available and can apply an intermediate coordinatization ourselves.</p>

            <!-- [[1, 1, 2, 5], [-2, -1, -3, -8], [2, 1, 4, 7], [-2, 1, 3, -7]] -->
            <!-- (-33/14, 1/2, 1, 1) -->
            <exercise>
                <statement>
                    <p>In <m>P_3</m>, the vector space of polynomials with degree at most <m>3</m>, find the vector representation of <m>p = x^{3} + x^{2} + \frac{1}{2} \, x - \frac{33}{14}</m> relative to the basis for <m>P_3</m>: <md>
                        <mrow>B = \{&amp;
                                    5x^{3} + 2x^{2} + x + 1,\,
                                   -8x^{3} - 3x^{2} - x - 2,</mrow>
                        <mrow>&amp; 7x^{3} + 4x^{2} + x + 2,\,
                                   -7x^{3} + 3x^{2} + x - 2\}</mrow>
                    </md>.</p>

                    <p>Hint:  Coordinatize with respect to the basis <m>\left\{1, x, x^2, x^3\right\}</m>.</p>
                </statement>
            </exercise>


            <sage><input>
            A = matrix(QQ, [[1, -2, 2, -2], [1, -1, 1, 1], [2, -3, 4, 3], [5, -8, 7, -7]])
            B = A.columns()
            B
            </input></sage>

            <p><c>B</c> is a basis, since <c>A</c> is nonsingular.</p>

            <sage><input>
            A.is_singular()
            </input></sage>

            <p>Now coordinatize <c>p</c>.</p>

            <sage><input>
            p = vector(QQ, [-33/14, 1/2, 1, 1])
            p
            </input></sage>

            <p>We'll get a coordinatization old-style.</p>

            <sage><input>
            aug = column_matrix(B + [p])
            r = aug.rref()
            r
            </input></sage>

            <p>Let's check to see if this is right and we can recover <c>p</c>.</p>

            <sage><input>
            soln = r.column(4)
            round_trip = sum([soln[i]*B[i] for i in range(4)])
            round_trip, round_trip == p
            </input></sage>
        </section>

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