Existence proof of subspace of projections from V to W Hi I am having some trouble with this proof. I don't know in which direction to start. I was thinking of using the inner product over C and applying the conditions, then showing that the subspace is linear and non empty. Is this a good way to go?
Let V be a finite-dimensional vector space over $\Bbb{C}$ and let T:V $\rightarrow$ V be a linear transformation. Suppose that T is idempotent ($T^2$ = T) and self-adjoint (T* = T). Prove that there exists a subspace W of V such that for all v $\in$ V, the vector T(v) is the projection of v onto W.
Edit: Can someone please give a formal proof so I can understand what is going on. Thanks.
 A: Foremost, show that $T^2 = T$ implies that $V = Null(T) \oplus Im(T)$ (when $V$ is finite-dimensional).
(This is relatively easy to do using the containment properties of null spaces and images of powers of T: You need 4 facts to show this:
$$\{0\} \subseteq Null(T) \subseteq Null(T^2) \subseteq  \ ... \ \subseteq Null(T^m) \subseteq \ ... $$ 
$$ V \supseteq Im(T) \supseteq Im(T^2) \supseteq \ ... \ Im(T^m) \supseteq  \ ...$$
You will also need that (for complex finite dimensional vector spaces): $V = Null(T^n) \oplus Im(T^n)$ (where $dim(V) = n$)
And that if $Null(T) = Null(T^2)$, then $Null(T) = Null(T^m) \ \forall m \geq 2$
If you don't know these facts, there is another way to go about proving that $T^2 = T$ implies $V = Null(T) \oplus Im(T)$ without these, but I personally like this way best.
For the rest of the proof:
We also know that $Null(T) = (Im(T^*))^\perp$.
Since T is self-adjoint, we know that $Im(T^*) = Im(T)$
Hence $Null(T) = (Im(T))^\perp$
As the comment suggested, consider $W = Im(T)$. The previous statement implies $Null(T) = W^\perp$
Then for any $v \in V$, we can write $v = n + w$, where $n \in Null(T) = W^\perp$ and $w \in ImT = W$ (because $V = Null(T) \oplus Im(T)$, as you should verify)
Since $w \in Im(T)$, write $ w = T(u)$ for some $u \in V$. 
Then $T(v) = T(n + w) = T(n + T(u)) = T^2(u) = T(u) = w$.
This is exactly the definition of an orthogonal projection map onto W.
