Reorganizing/Merging of Some Linear Algebra Lessons (MX2, MX3, and change of basis/GT5)

[jkostiuk]

I'd like to pitch a suggestion for re-organizing some content in Linear Algebra--original conversation started in #776

Context: Section GT5 of the activity book is change-of-basis. The file is in the main project, but I don't believe it's published as part of the main edition yet. My instructors and I taught with this in Fall and Spring and the lesson itself went well for us. Now that we're thinking about including this in the main edition, we want to take a step back and make sure the lesson is in the right place and that the other lessons around it flow.

My specific suggestions for reorganizing:

1. merge MX3 into MX2. Given the way we defined the inverse and inverse matrix, I think it follows naturally in that lesson that we can use the inverse matrix to solve a system of equations. then, in the MX2 checkit, we can have task 1 be "explain why this matrix isn't invertible" task 2 can be your reworked "find the inverse" and then we can add a short task 3 that's "use your inverse to solve the system". While this makes MX2 a little longer, the ideas are well-connected: we defined the inverse matrix/trafo to be "the unique solution to this equation" so it would make thematic sense for the lesson to end with "now we can easily compute this unique solution".
2. Then, GT5 becomes a new MX3.
3. If we do this, then I think we can add 2-3 activities to GT4 to do some version of diagonalization (i.e., after computing a basis of eigenvectors, use that to diagonalize a matrix).

What does the community think? (Note that change-of-basis cannot happen earlier than this suggestion without significant edits because it requires the matrix inverse).

Tagging some specific people, but all comments welcome here!
@siwelwerd @StevenClontz @jford1906 @jbunn3 @megancl13

[jkostiuk]

I'm happy to work on the edits myself if/when we have a plan that seems reasonable.

[siwelwerd]

At one point there was consideration to have diagonilization in the in-development module ON. See https://github.com/TeamBasedInquiryLearning/library/blob/main/source/linear-algebra/source/future-ON/05.ptx.bak for some code fragments.

[jkostiuk]

thanks for sharing drew. While it's housed in the ON module, the title and contents make it seem like the intention was to include it in GT.

Thinking ahead, the ON module is something I might take a crack at during sabbatical, while the edits I'm suggesting above are something i could see myself getting through in the next few days if we decide we want this to be in the next edition.

[siwelwerd]

One consideration that may not be obvious from just reading the activities: it was advantageous to have two easy wins for students late in the course in MX2 and MX3, both being relatively easy standards for students to learn. Not saying we have to keep it that way, but something to be weighed against the other considerations.

Otherwise this seems reasonable. My big gripe with diagonalization is that it is never well motivated, so I'd like to see a good motivation for it if we include it. The standard "we can easily compute powers of matrices if its diagonalizable" is unsatisfying at best to most students (and me).

[jbunn3]

Yeah, I think the idea of having diagonalization in GT4.

[siwelwerd]

thanks for sharing drew. While it's housed in the ON module, the title and contents make it seem like the intention was to include it in GT.

That file might very well date back to the first institute in 2021. Tacking it on to the GT module was definitely an idea at one point, but not sure if there was a compelling reason at the time.

I mostly mentioned it just to make sure we prevent duplication of efforts rather than to suggest ON was a better home than GT.

[jkostiuk]

Otherwise this seems reasonable. My big gripe with diagonalization is that it is never well motivated, so I'd like to see a good motivation for it if we include it. The standard "we can easily compute powers of matrices if its diagonalizable" is unsatisfying at best to most students (and me).

Here was my idea: the change-of-basis section we have already emphasizes that we can now work with a linear transformation from a different perspective, no longer tied to the standard basis. In change-of-basis section, this would mostly be "hey, we want the flexibility to choose a different basis because why wouldn't we want that freedom?" Then, the punchline in GT4 would be "hey, here is what happens if we choose a basis that is determined by the geometry of the transformation."

We can then foreshadow that, in most applications when we want to work with a different basis, there's usually a geometric reason underlying the choices that we make.

I think the matrix powers stuff could be a good math writing exploration but agree with you that it isn't very satisfying until we have an application for that. As a side note, though, we had a good problem set problem this year that used matrix powers to find a closed formula for a recurrence relation (a modification of the fibonacci sequence) based largely on these notes. I quite liked this illustration.

[jkostiuk]

One consideration that may not be obvious from just reading the activities: it was advantageous to have two easy wins for students late in the course in MX2 and MX3, both being relatively easy standards for students to learn. Not saying we have to keep it that way, but something to be weighed against the other considerations.

I think this is a good point, and I'be curious to read what others think. The origin of this discussion stemmed from @StevenClontz 's observation that the CheckIt problem for change-of-basis and current-MX3 were actually quite similar. The suggestion I had above is essentially adding one part to MX2 (multiply the inverse by a specific vector), and then new-MX3 is basically doing the same thing, but after recognizing that they can write down the inverse-change-of-basis matrix and then need to calculate the inverse, then multiply.

So the outcomes become a little meatier, but I'm tempted to think they'd still be on the lighter side compared to some of our more significant outcomes.

[StevenClontz]

I think this is the right way to go: "change of basis" is most directly an application of matrix inverses to solve a linear equaiton with a unique solution, so putting this there sounds good.

For backwards design purposes, I'll first make a PR with a modified MX2, MX3 that incorporates change of basis, then @jkostiuk (if he likes it) can reorganize activities to match.