# Eigenspace and dimensions of linear transformation in complex plane

Suppose V is a finite-dimensional complex vector space and T : V → V is a linear transformation which satisfies $$T^{2} = T$$. Prove that for any vector v $$\in$$ V , the vector T(v) is contained in the eigenspace $$E_{1}$$ of eigenvalue $$1$$ and v −T(v) is contained in the eigenspace $$E_{0}$$ of eigenvalue $$0$$. Also prove that T is diagonalizable, and compute the dimensions of $$E_{0}$$ and $$E_{1}$$ in terms of the rank of T.

From $$T^{2}-T=0$$, I could get $$\lambda=0$$ or $$\lambda= 1$$ but I do not know how to continue.

Hint: If $$v\in V$$, then $$v=\bigl(v-T(v)\bigr)+T(v)$$.