Let V be a real or complex vector space and let T L(V ) be such that T2 =I. Define M={vV |Tv=v}, N={vV |Tv=v}. Prove that M and N are subspaces of V...

Let V be a real or complex vector space and let T ∈ L(V ) be such that T2 =I.

Define

M={v∈V |Tv=v}, N={v∈V |Tv=−v}.

Prove that M and N are subspaces of V and that V = M ⊕ N.

Hint: For every vector v, we have

v = 1 (v + T v) + 1 (v − T v). 22