My own question. I am sure it has an answer. But my Linear Algebra is so embarrasingly awful I am not sure. I have a dislike to the Linear Algebra involving matrix computations and applications of that such as linear mappings from \mathbb{R}^n\mapsto\mathbb{R}^m. However, there is a side of Linear Algebra that I like, it is about general vector spaces over fields (and therefore no matrices). I was thinking whether there is a generalization of "diagonalization" in general:

Let V,U be (finite) vector spaces over a field F. Let \phi: V\mapsto U be a linear transformation. Is there such a notion as " \phi is a diagonalizable linear transformation".