I'm struggling with a question on linear operators and direct sums. If someone could possibly help me out here that would be great. The question is as follows:
Let $V$ be a vector space.
Let $U$ and $W$ be subspaces $V$, such that $ V=U⊕W $. Define $ T\colon V \to V $ by $T(v)=w$ where $v$ is written (uniquely) as $v=u+w$ with $u$ in $U$ and $w$ in $W$.
Show that $T$ is a linear operator, $U=\ker T$, $W=\mathrm{im}\,T$, and $T^2 = T$.
I'm not sure how to approach this question. I know i should start with what we know which is the definition of a direct sum where $V=U⊕W$ needs to fulfil the 2 conditions - 1) $V=U+W$ and 2) $U∩W={0} $
Thanks a lot in advance.