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(kx) = 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.")
(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(AT) = 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 − λI) = 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.