Let $φ: V→V$ be a linear map with $φ \circ φ = \text{id}$. If $U = \{v \in V | φ(v) = v\}, W = \{ v \in V | φ(v) = -v\}$. Prove $ V \cong U \oplus W$

Question

Let $φ: V→V$ be a linear map with $φ \circ φ = \text{id}$. If $U = \{v \in V \mid φ(v) = v\},\ W = \{ v \in V \mid φ(v) = -v\}$. Prove $ V \cong U \oplus W$

I can't find a way to express $v \in V$

$v = u + w, u \in U, w \in W$

Answer 1

Hint: Put $u=(v+\varphi(v))/2$ and $w=(v-\varphi(v))/2$.