AT3 Incorporate Vector Equations

[jkostiuk]

I reworked 3.3.5 and 3.3.10 to reinforce the translation of an equation involving a transformation to a vector equation

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[jkostiuk]

copy-pasting comments from an instructor so I don't lose them (haven't yet reflected on it):

_I feel like there could still be stated a Fact: the image is equal to the span of these column vectors. I scaffolded a warmup activity for my students today to try do this. It was something like:
Find a linear combination of T(e1),... that is equal to the vector T(1,3,-0.5,7) that's in the image. And explain why you know these things are equal, without having to actually do all of the computations. This shows everything in this span is a member of the image.
Find a vector in the domain that transforms to the linear combination 4*T(e1)+.... This explains why everything in the image is a member of this span.
I think my students internalized this too well and we had some disagreement about whether there was actually anything to prove here. I tried to highlight that these definitions, of image and span, are a priori totally unrelated, and there has to be an explanation of why they're connected.

There could maybe also be a Fact: the kernel is the solution set of this homogeneous vector equation. Just to spell out the connection between the linear transformation objects and the vector equation/linear system objects_

[StevenClontz]

I had the exact thoughts as your TAs when I did this today. Going to try to push a hotfix here to reflect my on the fly changes...

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[StevenClontz]

My students pointed out that in this specific example, the answer "infinitely-many solutions" is valid, because the set is dependent (there cannot be exactly one solution). I'm going to tweak the example to avoid this.

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[StevenClontz]

Apolgies for writing over your work rather than setting up a separate PR in my haste to make some changes between my classes. The latest commit reflects how I ended up pitching things in the second class (or the changes I wish I had made). What do you think?

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[jkostiuk]

@StevenClontz if you request a review from me, I can make some minor edits to the grammar for your subsequence review.

[StevenClontz]

I can't request your review for your own PR. But you can still suggest changes using the "Files changed" tab.

[jkostiuk]

y'know @StevenClontz, after being away from this for a long time, I think these activities could still use some work.

Specifics of any activities in AT3 aside, a core idea I would want my students to get comfortable with is the use of linear transformations in forming/solving equations. That is, I want students to consider questions/statements about equations of the form T(x)=w and be comfortable with translating these into question/statements about vector equations.

In this PR, activities 3.3.5/remark 3.3.6 i think hit the mark (for me), but 3.3.11 doesn't quite handle this. I'm going to draft a different PR so we don't lose this one and then I'm hoping you and I can sync up live this week to resolve?