Hint for problem II.3.4, part (a)
Assume that P+Q is a projection. We have to prove that the ranges of
P and Q are orthogonal.
To see successive hints, highlight the items below.
- Write the idempotent condition for
P+Q.
- This gives an equation involving P and
Q.
- Assume by contradiction that there is x in
ran(Q) such that
P x is nonzero. Plug the vector x into the equation found
earlier.
- Note that I+Q is invertible!