Shut up! Shut up and leave me alone with my books!
Dear [name redacted]:
So, I'm trying (mostly unsuccessfully) to stop being bitter, because I'm powerless to change anything, and so being bitter is a waste of time when I could be doing something useful instead, but I still don't understand how a good person like you can actually think our so-called educational system is actually a good idea. I can totally understand being practical and choosing to work within the system because it's all we've got; there's nothing wrong with selling out as long as you get a good price. If you think you're actually helping your students become better thinkers and writers, then that's great, and you should be praised for having more patience than me. But I don't understand how you can unambiguously say that this gargantuan soul-destroying engine of mediocrity deserves more tax money without at least displaying a little bit of uncertainty!
Say we have a linear transformation A and some nonzero vector v, and suppose that Av = λv for some scalar λ. This is a very special situation; we say that λ is an eigenvalue of A corresponding to the eigenvector v.
How can we find eigenvalues? Here's one criterion. If Av = λv for some unknown λ, we at least know that Av – λv equals the zero vector, which implies that the linear transformation (A – λI) maps v to zero. If (A – λI) maps v to zero, then it must have a nontrivial kernel, which is to say that it can't be invertible, and this happens exactly when its determinant is zero, because the determinant measures how the linear transformation distorts (signed) areas (volumes, 4-hypervolumes, &c.), so if the determinant is zero, it means you've lost a dimension; the space has been smashed infinitely thin. But det(A – λI) is a polynomial in λ, and so the roots of that polynomial are exactly the eigenvalues of A.
"Gee, Brain, what do you want to do tonight?"
"The same thing we do every night, Pinky—the Right Thing to Do Given Current Info and Preferences!"