(a) (i) State the rank-nullity theorem.

Let $U$ and $W$ be vector spaces. Write down the definition of their direct sum $U \oplus W$ and the inclusions $i: U \rightarrow U \oplus W, j: W \rightarrow U \oplus W$.

Now let $U$ and $W$ be subspaces of a vector space $V$. Define $l: U \cap W \rightarrow U \oplus W$ by $l(x)=i x-j x .$

Describe the quotient space $(U \oplus W) / \operatorname{Im}(l)$ as a subspace of $V$.

(ii) Let $V=\mathbb{R}^{5}$, and let $U$ be the subspace of $V$ spanned by the vectors

$$\left(\begin{array}{c} 1 \\ 2 \\ -1 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{c} -2 \\ 2 \\ 2 \\ 1 \\ -2 \end{array}\right)$$

and $W$ the subspace of $V$ spanned by the vectors

$$\left(\begin{array}{c} 3 \\ 2 \\ -3 \\ 1 \\ 3 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right),\left(\begin{array}{c} 1 \\ -4 \\ -1 \\ -2 \\ 1 \end{array}\right)$$

Determine the dimension of $U \cap W$.

(b) Let $A, B$ be complex $n$ by $n$ matrices with $\operatorname{rank}(B)=k$.

Show that $\operatorname{det}(A+t B)$ is a polynomial in $t$ of degree at most $k$.

Show that if $k=n$ the polynomial is of degree precisely $n$.

Give an example where $k \geqslant 1$ but this polynomial is zero.