Linear Algebra Done Right · Problem 6C.9

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthogonal-projections

Problem

Suppose

V

is finite-dimensional. Suppose

P \in \mathcal{L}(V)

is such that

P^2 = P

and every vector in null

P

is orthogonal to every vector in range

P

. Prove that there exists a subspace

U

V

such that

P = P_U