(a) Let $U, U^{\prime}$ be subspaces of a finite-dimensional vector space $V$. Prove that $\operatorname{dim}\left(U+U^{\prime}\right)=\operatorname{dim} U+\operatorname{dim} U^{\prime}-\operatorname{dim}\left(U \cap U^{\prime}\right) .$

(b) Let $V$ and $W$ be finite-dimensional vector spaces and let $\alpha$ and $\beta$ be linear maps from $V$ to $W$. Prove that

$$\operatorname{rank}(\alpha+\beta) \leqslant \operatorname{rank} \alpha+\operatorname{rank} \beta$$

(c) Deduce from this result that

$$\operatorname{rank}(\alpha+\beta) \geqslant|\operatorname{rank} \alpha-\operatorname{rank} \beta|$$

(d) Let $V=W=\mathbb{R}^{n}$ and suppose that $1 \leqslant r \leqslant s \leqslant n$. Exhibit linear maps $\alpha, \beta: V \rightarrow W$ such that $\operatorname{rank} \alpha=r, \operatorname{rank} \beta=s$ and $\operatorname{rank}(\alpha+\beta)=s-r$. Suppose that $r+s \geqslant n$. Exhibit linear maps $\alpha, \beta: V \rightarrow W$ such that $\operatorname{rank} \alpha=r, \operatorname{rank} \beta=s$ and $\operatorname{rank}(\alpha+\beta)=n$.