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IMC 1994 D1 P4

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March 6, 2017
IMClinear algebra

Problem Statement

Let α∈R\{0}\alpha\in\mathbb R\backslash \{ 0 \} and suppose that FF and GG are linear maps (operators) from Rn\mathbb R^n into Rn\mathbb R^n satisfying F∘G−G∘F=αFF\circ G - G\circ F=\alpha F.
a) Show that for all k∈Nk\in\mathbb N one has Fk∘G−G∘Fk=αkFkF^k\circ G-G\circ F^k=\alpha kF^k.
b) Show that there exists k≥1k\geq 1 such that Fk=0F^k=0.
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