Seems like I still don't get it, I think I am missing something important.
Let $V$ be an $n$ dimensional inner product space ($n \geq 1$), and $T\colon\mathbf{V}\to\mathbf{V}$ be a linear transformation such that:
- $T^2 = T$
- $||T(a)|| \leq ||a||$ for every vector $a$ in $\mathbf{V}$;
Prove that a subspace $U \subseteq V$ exist, such that $T$ is the orthogonal projection on $U$.
Now, I know these things:
- The fact hat $T^2 = T$ guarantees that $T$ is indeed a projection, so I need to prove that T is an orthogonal projection (I guess this is where $||T(a)|| \leq ||a||$ kicks in).
- To do this I can prove that:
- For every $v$ in $ImT^{\perp}$, $T(v) = 0$
- Alternatively, I can prove that for every $v$ in $ImT$ and $u$ in $KerT$, $(v,u)=0$.
- $T$ is self-adjoint (according to Wikipedia)
- The matrix $A = [T]_{E}$ when $E$ is an orthonormal basis, is hermitian (this is equivalent to the previous point).
- What else?
I've been thinking about it for quite some time now, and I'm pretty sure there is something big I'm missing, again. I just don't know how to use the data to prove any of these things.
Thanks!