Let T: V -> V, V is a finite-dimensional vector space

T^2 is identity operator.

lamda be a scalar.

The eigenspace V^(lamda) is the set of eigen-vectors of T with eigen-value lamda, together with zero.

Prove that V^(lamda) is a T-invariant subspace.

Prove that for all v in V, v - Tv is either an eigen-vector with eigen-value -1 or zero vector

Prove that V is direct sum of the eigenspaces V^(1) and V^(-1)

Please help