Alright, a question for all you non-Tommy-disciples out there (see, I was going to say apostles, but I resisted the tempation. You should be very proud of me
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i'm strongly in favor of starting with (3-d) parallelopiped volume. it at least gives a solid framework for explaining multilinearity, and why det of a non-full-rank matrix is 0. yes, there are plenty of fancier things to say (most of which i don't remember), but at the high school level, you should be careful about how deep you dive into abstractions (unless you've got a real standout student)
for matrix multiplication, you can then start by talking about multiplying by c*I, and then try to convince him that it's reasonable to think of determinant-1 matrices as volume-preserving operations (i don't have much of an idea of how to do that part for an hs student)
Myeah, I was thinking of that... though I wonder if dragging 3d geometry into this won't just make life more painful.
I'm still a fan of the 'matrices as operations that convert one vector into another' approach, just 'cause that's the only thing that made any semblance of sense back in middle school. But while that leads to a natural way of explaining matrix-by-vector and matrix-by-matrix multiplication, it doesn't give an obvious reason for the determinant.
"operations" are harder to visualize than parallelopipeds, if you want them to have any intuition. and intuition=good.
you don't need to drag abstruse stereometry formulas into it. start with a rectangular prism. then shear it along one dimension. explain why volume stays constant by analogy to rectangle/parallelogram. etc.
Yeah when i learned about determinants it was more like a measure of whether the matrix is degenerate or not. English description for degenerate is whether you can have two different lists (#giraffes and #monkeys) which produce the same requirement for apples and oranges.
Of course it is not really a good measure of matrix conditioning, but zero vs non-zero is pretty good for starters.
Volume is next thing -- a pretty simple derivation is that if you have two vectors (x1, y1) and (x2, y2) then the area of a triangle defined by these vectors is equal to full enclosing rectangle minus 3 side-way right triangle. is x1*y2 - 1/2(x1*y1) - 1/2(x2*y2) - 1/2 (x1-x2)*(y2-y1) = 1/2(x1*y2 - y2*x1)
The generalization to 3 dimensional space is similar (except that you have 4 right prisms.
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I'm still a fan of the 'matrices as operations that convert one vector into another' approach, just 'cause that's the only thing that made any semblance of sense back in middle school. But while that leads to a natural way of explaining matrix-by-vector and matrix-by-matrix multiplication, it doesn't give an obvious reason for the determinant.
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you don't need to drag abstruse stereometry formulas into it. start with a rectangular prism. then shear it along one dimension. explain why volume stays constant by analogy to rectangle/parallelogram. etc.
also, you were doing WHAT in middle school?
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Of course it is not really a good measure of matrix conditioning, but zero vs non-zero is pretty good for starters.
Volume is next thing -- a pretty simple derivation is that if you have two vectors (x1, y1) and (x2, y2) then the area of a triangle defined by these vectors is equal to full enclosing rectangle minus 3 side-way right triangle.
is x1*y2 - 1/2(x1*y1) - 1/2(x2*y2) - 1/2 (x1-x2)*(y2-y1) = 1/2(x1*y2 - y2*x1)
The generalization to 3 dimensional space is similar (except that you have 4 right prisms.
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