Linear Algebra Done Right · Problem 3D.17

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · invertible-linear-maps

Problem

Suppose VV is finite-dimensional and S∈L(V)S \in \mathcal{L}(V). Define A∈L(L(V))\mathcal{A} \in \mathcal{L}(\mathcal{L}(V)) by A(T)=ST\mathcal{A}(T) = ST for T∈L(V)T \in \mathcal{L}(V). (a) Show that dim⁡null⁡A=(dim⁡V)(dim⁡null⁡S)\dim \operatorname{null} \mathcal{A} = (\dim V)(\dim \operatorname{null} S). (b) Show that dim⁡range⁡A=(dim⁡V)(dim⁡range⁡S)\dim \operatorname{range} \mathcal{A} = (\dim V)(\dim \operatorname{range} S).

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