Maths tutoring has now switched over from analysis to linear algebra. This may force me to finally actually understand linear algebra.
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(It's not exactly true that I don't understand linear algebra. At a conceptual level I understand it very well, but I struggle to put that into practice for a variety of reasons)
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I think a lot of the problem here is that linear algebra is about 10% useful abstractions, 60% notation and 30% clever algorithms operating on that notation, but the notation is: a) Poorly explained. b) Often means multiple different things or is implicitly punned.
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I remember a supervision (tutorial) back at university where a supervisor got really exasperated us for pointing out that something in an equation was actually a 1x1 matrix and not a scalar and that this meant that the operations being done were technically invalid.
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This matters because if you pun scalars and 1x1 matrixes then a lot of expressions no longer become associative because the reassociated version wouldn't match up. e.g. (x^t y) z is valid if you pun x^t y as a scalar, but x^t (y z) is an invalid multiplication.
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But this punning is rife and unacknowledged in how people write linear algebra, and to some extent that's fine, but it means that your notation is full of context-dependent interpretations of the meaning of expressions.
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There’s a funny rant by someone (John D Cook maybe?) about how every symbol in probability theory is P, and you often have an equation with five different Ps which, to a computer scientist, are all of glaringly different types, but somehow statisticians don’t notice it
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