Let $U$ and $V$ be finite dimensional vector spaces and $\alpha: U \rightarrow V$ a linear map. Suppose $W$ is a subspace of $U$. Prove that

$$r(\alpha) \geqslant r\left(\left.\alpha\right|_{W}\right) \geqslant r(\alpha)-\operatorname{dim}(U)+\operatorname{dim}(W)$$

where $r(\alpha)$ denotes the rank of $\alpha$ and $\left.\alpha\right|_{W}$ denotes the restriction of $\alpha$ to $W$. Give examples showing that each inequality can be both a strict inequality and an equality.