fuzzy test for row equivalency

[Alex-Jordan]

This adds an isREQ method for Matrix objects. Used like $A->isREQ($B).

This is a "fuzzy " check that avoids both machine rounding issues and student rounding issues (for example if they enter something like 0.3333 instead of 1/3). Instead of reducing each matrix to RREF (which could introduce extraneous pivot positions from machine rounding), this approach compares the two row spaces. If each row from one matrix is "close enough" to the other matrix's row space, then the algorithm effectively declares that row to be in the other matrix's row space. "Close enough" is based on normalizing nonzero rows to length 1, and then the usual fuzzy tolerances for comparing that normalized row and its projection onto the other matrix's row space.

For example, Matrix([ 3, 1 ], [ 1, 1 / 3 ]) and Matrix([ 1, 0.3333 ], [ 0, 0 ]) will be considered row equivalent.

But Matrix([ 3, 1 ], [ 1, 1 / 3 ]) and Matrix([ 1, 0.33 ], [ 0, 0 ]) will not be.

And of course Matrix([ 3, 1 ], [ 1, 1 / 3 ]) and Matrix([ 1, 0], [ 0, 1 ]) will not be.

For questions that ask a student to completely reduce $A, a checker could check $student->isRREF() && $student->isREQ($A).

[Alex-Jordan]

I updated the top comment here, partly to fix typos, but also to explain a little more about what the comparison actually does.

In another thread, @somiaj and I were talking about actually implementing an RREF algorithm. We could do that too, I just couldn't think of a way to avoid extraneous pivot positions popping up from machine rounding. That situation would create "false negatives" where two matrices that should be considered row equivalent would not be. But here, the algorithm goes the other way. It may create "false positives" because of the fuzzy tolerances.