Powerful applications of linear algebra?

Question

I'd like to see some neat, elegant applications of linear algebra. I'm a undergraduate student but I don't want to prevent people from posting things just because I won't understand them, but if it's undergraduate level even better.

Examples:

Cayley–Bacharach theorem

Every element in a finite extension of a field is algebric

Radon's theorem

Answer 1

In differential topology we discuss tangent spaces to smooth manifolds- it turns out that these tangent spaces are vector spaces essentially "sitting" on top of manifolds. Their dimension matches the dimension of the manifold.

A lot of properties crucial to classifying manifolds boil down to the action of linear maps which transform the tangent spaces of manifolds.

If $X$ and $Y$ are smooth manifolds and $f:X\to Y$, then $T_x(X)$ is the tangent space at the point $x$, and $T_{f(x)}(Y)$ is the tangent space of the image point of $x$ under $f$. The derivative of $f$ at $x, df_x$ is a linear map of vector spaces: that is $$ df_x:T_x(X)\to T_{f(x)}(Y).$$ Hopefully this is a sufficiently interesting example. Let me directly quote from an answer given by Qiaochu Yuan on Quora:

"Linear algebra intersects every other field of mathematics, all the time, everywhere. It's not much of an exaggeration to claim that the only part of mathematics that mathematicians really understand is linear algebra, and everything else is us trying to figure other stuff out by reducing it to linear algebra."

https://www.quora.com/What-are-some-important-points-of-intersection-between-linear-algebra-and-other-branches-of-mathematics

Answer 2

One of the most powerful applications that I know about is finding "main axis" moment of inertia using matrices diagonalization.

Also there are a lot of linear operators which are constantly used in mathematics/physics (derivative of finite polynomial for example.) and knowing linear algebra just deepens the understanding...