Is this problem incomplete? I don't know how to proceed! If it's incomplete, please let me know what details need to be added.
Let $V=U\oplus W$. Let $P_W:V\to W$ be the canonical projection and $R_W:V\to V$ be the reflection w.r.t $W$, i.e. it does $w+u\mapsto w-u$. Compute the minimal polynomials of $P_W$ and $R_W$.
Thank you!
Let $\{u_1,\ldots,u_k\}$ be a basis of $U$ and let $\{w_1,\ldots,w_l\}$ be a basis of $W$. Then $B=\{u_1,\ldots,u_k,w_1,\ldots,w_l\}$ is a basis of $V$. Besides, $P_W(u_j)=0$ for each $j\in\{1,2,\ldots,k\}$ and $P_W(w_i)=w_i$ for each $i\in\{1,2,\ldots,l\}$. So, the matrix of $P_W$ with respect to $B$ is a diagonal matrix such that the first $k$ entries of the main diagonal are all equal to $0$ and the remaining ones are all equal to $1$. So, the characteristic polynomial fo $P_W$ is$$(-\lambda)^n(1-\lambda)^l=(-1)^{\dim V}\lambda^n(\lambda-1)^l.$$Can you deal with $R_W$ now?