Let V be a finite dimensional vector space and $T \in L(V)$. Show that
a) if T^2 = T, then T is diagonalisable with eigenvalues 0 and 1, V = V1⊕V2 for some subspaces V1, V2 such that T is the projection on V1 along V2.
b) if T^2 = I (identity matrix), then T is diagonalisable with eighenvalues ±1, V = V1 ⊕ V2 for some subspaces V1, V2 such that T is the symmetry about V1 along V2.
I have shown how T is diagonalisable with eighenvalues 0 and 1 and T is diagonalisable with eighenvalues ±1 for a) and b). But I am stuck on the second part for both a) and b).
I know that "projection" in a) means that "T maps every v written uniquely as $v_1+v_2$ with $v_i$ $\in$ $V_i$ to Tv = $v_1$" and "symmetry" in b) means that "T maps every v written uniquely as $v_1+v_2$ with $v_i$ $\in$ $V_i$ to Tv = $v_1$-$v_2$." But I am trouble on showing them--are they related to the eigenvalues part i proved or not?