She learned Quantum Mechanics without Linear Algebra, dudes. Quantum mechanics

*is*linear algebra (in funny hats). But she'd seen vectors and dot products and determinants and the like before; she just hadn't had a formal course in them.

So I took it upon myself to teach her a nearly four-hour crash course in linear algebra, which took the second half of the day.

The first two hours were devoted to some review of

(1) What linearity is [ f(

__x__+

__y__) = f(

__x__) + f(

__y__), f(k

__x__) = k f(

__x__). ]

(2) Why every linear function can be characterized as a matrix.

(3) How to do matrix multiplication.

(4) Coordinate systems.

(5) A little bit of matrix inversion (especially the 2x2 case), but not the explicit case of solving linear systems.

(6) Questions, some problems.

Coffee break! I illustrated Scott Aaronson's "Proof by Pizza" and explained calculus in under a minute to a crowd of undergrad physics majors. I got a round of applause.

The second two hours covered:

(1) Eigenvectors. ("Making f(x + y) = f(x) + f(y) even simpler.")

(2) Eigenvalues.

(3) Eigenvalues in Quantum Mechanics.

(4) Matrices as seen from their diagonalizing coordinates.

(5) The determinant of a diagonal matrix, and the physical interpretation of it.

(6) Determinants in general, det(AB) = det(A) det(B), det(A

^{T}) = det(A).

(7) Why, if

**M**

__v__=

__0__for a nontrivial

__v__, then det

**M**= 0. (just diagonalize

**M**).

(8) How to actually find eigenvalues [ derivation of det(

**M**− λ

**I**) = 0. ]

(9) Finding an eigenvector with a given eigenvalue.

It sounds like more, but I didn't have to talk at great length about how to do determinants with Josina -- she understood from the start. She just didn't know why they were important and what they physically meant, so I tried to show her.

All in all, I thought it was a pretty good crash course.

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