PROTEUS RREF (Q5)

[rbeezer]

From @twjudson, cleaved off #151

[rbeezer]

      <exercise>
        <title>Calculating row reduced echelon form</title>
        <statement>
            <p>Arrange the blocks in order to show how the algorithm works for calculating the row reduced echelon form of the matrix
                          <me>
              A =
                  \begin{bmatrix}
                      3 &amp; 4 &amp; -1 &amp; 2 \\
                      1 &amp; -2 &amp; 3 &amp; 1 \\
                      0 &amp; 10 &amp; -10 &amp; -1   
                  \end{bmatrix}.
        </me>
            </p>
        </statement>
        <blocks>
            <block order="1" correct="yes">
                <p><me>
                  A =
                      \begin{bmatrix}
                          3 &amp; 4 &amp; -1 &amp; 2 \\
                          1 &amp; -2 &amp; 3 &amp; 1 \\
                          0 &amp; 10 &amp; -10 &amp; -1   
                      \end{bmatrix}.
                </me></p>
            </block>
            <block order="2" correct="yes">
                <p><me>
                  A =
                      \begin{bmatrix}
                          1 &amp; -2 &amp; 3 &amp; 1 \\
                          3 &amp; 4 &amp; -1 &amp; 2 \\
                          0 &amp; 10 &amp; -10 &amp; -1   
                      \end{bmatrix}.
                </me></p>
            </block>
            <block order="3" correct="yes">
                <p><me>
                  A =
                      \begin{bmatrix}
                          1 &amp; -2 &amp; 3 &amp; 1 \\
                          0 &amp; 10 &amp; -10 &amp; -1 \\
                          0 &amp; 10 &amp; -10 &amp; -1   
                      \end{bmatrix}.
                </me></p>
            </block>
            <block order="4" correct="yes">
                <p><me>
                  A =
                      \begin{bmatrix}
                          1 &amp; -2 &amp; 3 &amp; 1 \\
                          0 &amp; 1 &amp; -1 &amp; -1/10 \\
                          0 &amp; 10 &amp; -10 &amp; -1   
                      \end{bmatrix}.
                </me></p>
            </block>
            <block order="5" correct="yes">
                <p><me>
                  A =
                      \begin{bmatrix}
                          1 &amp; -2 &amp; 3 &amp; 1 \\
                          0 &amp; 1 &amp; -1 &amp; -1/10 \\
                          0 &amp; 0 &amp; 0 &amp; 0  
                      \end{bmatrix}.
                </me></p>
            </block>
            <block order="6" correct="yes">
                <p><me>
                  A =
                      \begin{bmatrix}
                          1 &amp; 0 &amp;-1 &amp; 1/5 \\
                          0 &amp; 1 &amp; -1 &amp; -1/10 \\
                          0 &amp; 0 &amp; 0 &amp; 0  
                      \end{bmatrix}.
                </me></p>
            </block>                 
        </blocks>
    </exercise>

[rbeezer]

Added in at 5d6815d

[rbeezer]

Some markup comments:

• @correct is likely only necessary to mark the no cases, I think yes is the default.
• @order is how the blocks will go in a static format. Unlikely you want the correct order for that?

Content comments:

• Pretty early in writing FCLA I started to create examples that only had integer entries. Not rationals. ;-)

I think this makes for a great Parsons problem. It forces you to realize you just need to see the evolution of the pivot columns, and none of the other arithmetic.

[rbeezer]

Forgot to say:

The problem needs a @component, @xml:id, and @label. See the commit with the added version.

And I just adjusted the @label, which had a typo (at abb26e2).

[rbeezer]

More ideas.

• Maybe a period in the #p for each block is not really adding much?
• Instead of "the algorithm", how about a link back to the relevant content in the book? (I'm sort of curious how that'll work technically.)
• I never say "works" in class. Maybe something like "the steps that compute a matrix that is row-equivalent to A and in reduced row-echelon form".

[dfrankdt]

Might be useful to include some more detailed instruction to ensure the first block is the matrix A itself.

[twjudson]

[dfrankdt]

The code could probably use a link to the algorithm in the book

[Tanaquil18]

We might want to consider whether having students focus on the algorithm is a good use of their cognitive load. Most of the time, we tell students to use Sage to find the rref, so do we care whether they know the steps of the algorithm as defined in FCLA?

[rbeezer]

I think it is very important to have students mechanically row-reduce matrices initially. I put one such question on the first (of seven) exam.

Why?

Row-equivalent matrices have equal row spaces. So the nonzero rows of RREF are a basis of the row space. Mechanically forming linear combinations of rows (row operations) helps to drive home this critical observation.

[Tanaquil18]

I agree that doing row operations by hand is important. The thing that gives me pause is that "the algorithm" is not emphasized very much, and so I don't like asking students to follow an exact set of steps with one possible ordering, especially as a reading question before coming to class. My students learned a different algorithm in their 1-credit course they take before linear algebra (all the forward steps first and then all the backward steps) and they also wouldn't have swapped the first two rows first unless they were trying to avoid calculational errors. If I'm being fully honest, I'm still not sure I understand what in your pseudocode or examples tells you which rows to swap or when not to swap. I get if there's a 0 in a pivot position, but you usually seem to swap to put a 1 or -1 in a pivot position if you're able to and otherwise don't swap and just go straight to scaling. Is that right?

Can we get at making them doing row operations in a way other than a Parson's problem? A short answer asking them to report which row operations they did to find the equivalent matrix in rref? Give them a list of row operations to perform and ask students to put the corresponding matrices that result from doing each step in order? Actually, I have written one like before. Reading Question 2: Parsons Problem, Sequence of Row Operations
https://tanaquil18.github.io/matrixcalculations/sec-row-ops.html#parsons-row-ops-sequence