Let V be a vector space over R. Show that if T is a linear operator on V, the following conditions are equivalent 1. T 2 = I V , where I V is the identity in V 2. V is the direct sum of the kernel of...

Let V be a vector space over R. Show that if T is a linear operator on V, the
following conditions are equivalent
1. T²
= I_V, where I_V
is the identity in V
2. V is the direct sum of the kernel of (T - I_V) and the kernel of (T + I_V).
3. There are W and X subspaces of V such that V = W ⊕ X and T (w + x) = w - x, for all w ∈ W, x ∈ X