Show If T is idempotent, then there exists a basis {v1,...,vn} for V such that Tvi = vi for i = 1,...,ρ(T), and Tvi = 0 for i =ρ(T)+1,...,n.

Question

Consider a vector space V with dimV = n and a transformation T ∈ L(V,V ) (a) Show that if T is idempotent, then there exists a basis {v1,...,vn} for V such that Tvi = vi for i = 1,...,ρ(T), and Tvi = 0 for i =ρ(T)+1,...,n.

This is a problem from my Linear Algebra class. I thought we could probably find such a basis in order to prove existence. Is that possible? What is the right way to go about this question?

Answer 1

Hint: Take a basis $\{v_1,\dots,v_{\rho(T)}\}$ of the range of $T$ and a basis of the null space of $T$; how many elements should this basis have?