Linear Algebra Done Right · Problem 6C.6

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthogonal-projections

Problem

Suppose

V

is finite-dimensional and

T \in \mathcal{L}(V, W)

. Show that

T = T P_{(\operatorname{null} T)^\perp} = P_{\operatorname{range} T} T.