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 $\displaystyle \mathbb{R}^n\mapsto\mathbb{R}^m$. 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 $\displaystyle V,U$ be (finite) vector spaces over a field $\displaystyle F$. Let $\displaystyle \phi: V\mapsto U$ be a linear transformation. Is there such a notion as "$\displaystyle \phi$ is a diagonalizable linear transformation".