orthogonal projectors, and their Nullspace and Range

Feb 2009
98
0
Let P and Q be 2 orthogonal projectors. We prove that if Range(P) and Null(Q) are linearly independent, and Range(Q) and Null(P) are linearly independent, then \(\displaystyle \left\| P-Q\right\|_{2} < 1\)

Does Range(P) and Null(Q) are linearly independent mean that Range(P) \(\displaystyle \cap \) Null(Q) \(\displaystyle \neq \left\{0 \right\}\)?
If yes, how do you use this fact to prove the problem?