I am using Evar Nering's and Bernard Kolman's linear algebra texts.
"automorphisms are the only linear transformations with inverses." Nering.
"Theorem 6.7: Let $L:V\rightarrow W$ be a linear transformation. Then $L$ is invertible if and only if $L$ is one-to-one and onto." Kolman.
Nering did not provide a proof. My issue is I think not all one-to-one and onto linear transformations are automorphisms. Am I missing something here?
The statement “automorphisms are the only linear transformations with inverse” is about automorphisms, but theorem 6.7 isn't. You seem to be suggesting that they are in conflict, but they are about different things. But, yes, you are right when you assert that not all one-to-one and onto linear transformations (between possibly distinct vector spaces) are automorphisms. And… ?