RREF.xml

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

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

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        <frontmatter>
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        <p>To specify a matrix, first specify that your entries are rational numbers, <c>QQ</c> in Sage parlance.  This is extremely important, though an explanation is best saved for later.  Then the number of rows and columns.  Follow this with a list of rows for the matrix, where each row is another list.  We use square brackets, <c>[, ]</c> to organize lists.  To actually see <c>A</c> we just write it as the last line.</p>

        <sage>
            <input>
            A = matrix(QQ, 3, 4, [[-1,  1,  1,  1],
                                  [ 2, -1, -2, -3],
                                  [-2,  2,  1, -1]])
            A
            </input>
        </sage>

        <p>Then use the <c>.rref()</c> method to compute the reduced row-echelon form.</p>

        <exercise>
            <statement>
                <p>Use a new cell, enter <c>A.rref()</c>, and execute the cell.</p>
            </statement>
        </exercise>

        <sage />

        <p>Here is a larger example.</p>

        <sage>
            <input>
            B = matrix(QQ, [[ 0,  0, -1,  3,  3, -1,  2, -1],
                            [ 1, -5,  0, -4,  5, -2,  4,  4],
                            [-1,  5, -1,  7, -2,  2, -5, -7],
                            [-1,  5,  1,  1, -8,  3, -6, -3]])
            B
            </input>
        </sage>

        <exercise>
            <statement>
                <p>As above, use a new cell to compute the reduced row-echelon form of <c>B</c>.</p>
            </statement>
        </exercise>

        <sage />

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    </article>

</pretext>