Hi all, I was studying for a Math test until this question stumped me:

Let $\displaystyle V $ be an inner product space and $\displaystyle T : V \rightarrow V $ be a self-adjoint linear transformation such that $\displaystyle T^{2} = T $.

a) Show that all eigenvalues of $\displaystyle T $ are either 0 or 1.

b) Describe the eigenspaces of $\displaystyle T $ in terms of the kernel of $\displaystyle T $, the range of $\displaystyle T $ and $\displaystyle V $.

So for question a) I know that a self-adjoint linear transformation means $\displaystyle <T(u)|v> = <u|T(v)> \forall u, v \in V $ and $\displaystyle T(u) = \lambda u $ where $\displaystyle \lambda $ is a scalar but I don't know how to use these to solve the question...

As for b) I know that Nullity T + Rank T = Dim V which is equivalent to dim(ker T) + dim(im T) = dim V ...but I guess I can't solve this until I know how to do question a)...

Any help would be greatly appreciated.