Are There Infinite Isomorphisms?

Question

Is this true:

If $V$ and $W$ are isomorphic spaces, where $\dim(V)>0$ and $\dim(W)>0$, then there are infinite isomorphisms $L\colon V \to W$.

Answer 1

Three things:

• If $\dim V < \infty$ and the field is finite, we have that $V$ and $W$ are finite sets, hence admit only finitely many arbitrary maps between them.

• If the field is infinite, multiplication by all elements of $K \setminus \{0\}$ gives rise to infinitely many isomorphisms, once you have one given isomorphism.

• If the dimension of $V$ is infinite - say $X$ is a basis - we have $\operatorname{Sym}(X) \subset \operatorname{Aut}_K(V)$. The fact $|\operatorname{Sym}(X)|=\infty$ gives rise to infinitely many isomorphisms, once you have one given isomorphism. The size of the field does not matter for that argument.