# Thread: Nulls and Ranges

1. ## Nulls and Ranges

I got a question here that states:

Suppose that p belongs to L(V) and P² = P, show that V = nullP is a direct sum of rangeP.

Could anyone give me a rough idea on how to approach this question? Thanks for any help.

2. Originally Posted by GreenDay14
I got a question here that states:

Suppose that p belongs to L(V) and P² = P, show that V = nullP is a direct sum of rangeP.

Could anyone give me a rough idea on how to approach this question? Thanks for any help.
For any x in V, x = Px + (x–Px). Notice that Px is in ran(P), and x–Px is in null(P).

To complete the proof, you also have to show that $\text{ran}(P)\cap\text{null}(P) = \{0\}$. So suppose that y = Px and that Py = 0. Can you deduce that y = 0?