Internal sum of Eigenspace Let V be a $\mathbb{R}-$Vectorspace of arbitrary dimension and $f\in End(V)$ (Set of all endomorphisms)
show that if $f \circ f = f$ then V = E(f,1) + E(f,0).
So that means we want to prove that V = fixpoints of f + ker(f)? And f is its on identity but I can't seem to make a connection here.
Thanks.

$f\circ f=f$ implies for $x\in V$ that $f(f(x))=f(x)$, so $Im f$ consists of fixpoints. And for $x=f(x)+(x-f(x))$ You have $f(x-f(x))=f(x)-f(f(x))=f(x)-f(x)=0$ and thus $x-f(x)\in Kern(f)$. Of course a fixpoint in the kernel must be $0$. Thus $V=Im(f)\oplus ker(f)=E(f,1)\oplus E(f,0)$ .
