Linear Algebra Done Right · Problem 7A.7

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · adjoints

Problem

Prove that if T∈L(V,W)T \in \mathcal{L}(V, W), then (a) dim⁡null⁡T∗=dim⁡null⁡T+dim⁡W−dim⁡V\dim \operatorname{null} T^* = \dim \operatorname{null} T + \dim W - \dim V; (b) dim⁡range⁡T∗=dim⁡range⁡T\dim \operatorname{range} T^* = \dim \operatorname{range} T.

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