(1) Let $V$ be a finite-dimensional vector space and let $T: V \rightarrow V$ be a non-zero endomorphism of $V$. If $\operatorname{ker}(T)=\operatorname{im}(T)$ show that the dimension of $V$ is an even integer. Find the minimal polynomial of $T$. [You may assume the rank-nullity theorem.]

(2) Let $A_{i}, 1 \leqslant i \leqslant 3$, be non-zero subspaces of a vector space $V$ with the property that

$$V=A_{1} \oplus A_{2}=A_{2} \oplus A_{3}=A_{1} \oplus A_{3} .$$

Show that there is a 2-dimensional subspace $W \subset V$ for which all the $W \cap A_{i}$ are one-dimensional.