Thread: Prove that eigenspace is a T-invariant subspace

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

Originally Posted by mathbeginner

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

Let be the eigenspace with eigenvalue lambda. Then if
then



Notice that both and

w is an eigenvector if and

Calculate



This should get you started.