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?

[siwelwerd]

Is this still needed now that #901 has been merged?

[jkostiuk]

good catch, no it's not. do I just "close it" then? or is there a different process to delete it?

[siwelwerd]

Nope, just hit "Close pull request". We probably don't want it deleted (even if we could) since this conversation informed your work on #901. Closing keeps it where we can find it if we go looking for it, but the default is for the PR list to only show open PRs, so it will be out of sight unless we ever need to revisit this.

[jkostiuk]

sweet, thanks.