[tgb]

And they exist in situations where determinants are difficult or impossible to define! Infinite dimensional vector spaces can still have transformations with eigenvectors but you generally can't define a determinant coherently for them (certainly they might have infinitely many eigenvalues and if the determinant is the product, then you have convergence problems). A classic example is that the standard Gaussian distribution is an eigenvector of the Fourier transform.

[in9]

You just blew my mind with this comment. I'm a statistics undergrad student and I really need to up my math knowledge. Where is this fact from? Functional analysis?

[tgb]

Yeah that would be a good course to learn it - functional analysis is basically linear algebra on infinite dimensional vector spaces. (Though that sells it a little short - moving to infinite dimensional vector spaces requires incorporating some topology too.)