My version was: the course boils down to just two things: anything that is linear behaves like matrix multiplication; eigenvectors are the ones that don’t change direction.
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The hardest part probably is the structure. When I think about linear algebra, I have a family of densely interconnected concepts to play with; but when I teach I have to explain one thing at a time. So I can't explain how I really think about it until the last week of class.
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That’s really interesting and makes sense… I want to say “well, can’t you go top-down and explain the big picture first?”, but I can offer convincing objections to that approach…
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Yes, I'd agree with that. It still surprises me a bit how hard it is to explain things that are perfectly clear in my head---clear enough I can use them fluently and easily and consistently. But not clear enough, apparently, that I can always tell my students how they work.
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But there's a reason that "you don't really understand the material until you've taught it" is proverbial.
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