[jeffwass]

Linear algebra (and most calculus) didn't really click for me until I needed to apply it to physics problems.

The general idea started to make sense in mechanics class when I could see matrices are convenient shorthand for solving multiple variables at once, and which behave like 'regular' variables when trying to manipulate them algebraically.

Eigenvalues and eigenvectors didn't make sense until quantum mechanics, where your 'operator' is effectively a matrix and your 'state' is a vector. The allowed observed values are the eigenvalues, and your final state after measurement is the corresponding eigenvector. Wave functions are then an extension of this from discrete space (useful for modelling spins for instance) to the continuum of position space.

[Steuard]

I love eigenvectors. For me, the rest of linear felt like a bunch of rote rules and algorithms for figuring out vaguely-interesting stuff. And then suddenly I find out that there's this amazingly non-obvious but remarkably powerful structure hidden inside matrices that I'd never even been aware of before.

Quantum mechanics is a huge application of eigenvectors, but I also really enjoy things like the "moment of inertia tensor", whose eigenvectors are the natural axes of rotation. Or better still, coupled oscillators, where the eigenvectors give "normal modes" of vibration. (And if you look at coupled first-order differential equations, eigenvectors can tell you all sorts of things about "trajectories" of the solutions. There are great applications of that to things like population dynamics in biology.)