Show that P is a projection

Question

Let $ V$ be a $\mathbb{K}$ - vector space and let $U_1, U_2$ be linear subspaces of $V $ with $V=U_1\oplus U_2$. Futhermore let the linear map $P: V \to V $ be defined by $ P(v)=u_1$ for $v=u_1+u_2 \in V,u_i \in U_i $.

I need to show that $P$ is a projection and determine $Im(P)$ and $ Ker(P)$.

I don't really know how to do that. However as far as I know $P: V \to V $ is a projection if $P\circ P=P. $

So help would be really appreciated.

Answer 1

$\forall v = u_1 +u_2 \in V$,

$$(P\circ P)(v) = P(P(v)) = P(u_1) = ?$$

Now to find $Ker P$, remember that $Ker P = \{ v\in V, P(v) = 0 \}$. If you write $v = u_1 + u_2$, how can $P(v) = 0 $?