Proof that $f$ is an isomorphism if and only if $f$ carries a basis to a basis.

Question

$f$ is an isomorphism if and only if $f$ carries a basis to a basis.

This was stated in my linear algebra textbook without justification. I was wondering if people could please take the time to clarify this by proving that it is true.

Answer 1

If $f : V \rightarrow W$ is an isomorphism, then $V$ and $W$ has same dimension. By this its enough to show that the image of the basis for $V$ are linear independent. Conversely if $f$ carries basis to basis, then their dimensions are equal. So $f$ must be an isomorphism.

Answer 2

Hints:

1) Prove that a linear transformation $\;f:V\to V\;$ is $\,1-1\,$ (injective) iff $\;\ker f=\{0\}\;$, and that it is onto (surjective) iff it carries a generating set to a generating set.

2) if $\;\dim V<\infty\;$ , then $\;f:V\to V\;$ is $\;1-1\;$ iff it is onto iff is bijective.