Say V is a finite dimensional vector space, and $T_1: V \to\ V$ and $T_2: V \to\ V$ are linear transformations. Assume that $T_1$ is one to one and $T_2$ is onto.
I am uncertain if my proof that $T_1T_2$ is an isomorphism is valid.
If you don't want to look at the picture - Is it necessary to show that $T_1$ and $T_2$ are each isomorphisms themselves by Rank-Nullity?
Here is an image of my work: