Problem 5

The analogue of a group/ring homomorphism for vector spaces is called a linear transformation. If $V$ and $W$ are vector spaces over a field $F$, then the map $T:V\to W$ is a linear transformation if it preserves vector addition and scalar multiplication, that is,

for all $\mathbf{u},\mathbf{v}\in V$ and $\alpha\in F$.

To prove that two vector spaces are isomorphic, one needs to find a bijective linear transformation.

With this information, prove 1.35.