Linear Algebra Done Right · Problem 6C.20

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · pseudoinverse

Problem

Suppose VV is finite-dimensional and T∈L(V,W)T \in \mathcal{L}(V, W). Show that null⁡T†=(range⁡T)⊥andrange⁡T†=(null⁡T)⊥.\operatorname{null} T^\dagger = (\operatorname{range} T)^\perp \quad \text{and} \quad \operatorname{range} T^\dagger = (\operatorname{null} T)^\perp.

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