I am confused of the following:
Suppose that there is an linear transformation $T : V \rightarrow W$ that is invertible. To prove that $T$ is one-to-one and onto, the proof that I saw says the following:
Let $\mathbb{v}$ be in the kernel of $T$. Then, $T^{-1}(T(\mathbb{v})) = T^{-1}(\mathbb{0}) \rightarrow I(\mathbb{v}) =0$
I am not sure how one can reach the conclusion of $I(\mathbb{v}) =0$.