(All of this over finite dimensional vector spaces) In Artin, Algebra, Chapter 4, section 3., the author says "When we are dealing with linear operators. It is clear that we want to pick only one basis for $V$". Why is it clear? In the page before, it was proved that for any linear transformation you can choose basis in domain and codomain so it's matrix takes the block form $\begin{bmatrix}
I & 0\\
0 & 0
\end{bmatrix}$ where $I$ is the identity matrix. This at first sounds very desirable to work with. If we use the same basis, the best we can do is Jordan Canonical form, which is nowhere as simple. So why restrict ourselves to use the same basis? It doesn't seem "clear" at all.