Exercise 3.D.8

Proof. Let $w_1,\dots ,w_n$ be a basis of $W$ and $v_1,\dots ,v_n$ an inverse image of this basis when $T$ is applied. Because $w_1,\dots ,w_n$ is linearly independent, it follows that $v_1,\dots ,v_n$ is also linearly independent (you can check it).

Define $U$ by

It easy to see that $T{\text{|}}_U$ is both injective and surjective. Hence $T{\text{|}}_U$ is an isomorphism of $U$ onto $W$. □