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Question: let v be a vector space over a eld f...

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Let V be a vector space over a field F. We say a linear map T : V→V is nilpotent if there exists a positive integer m such that Tm = 0 (that is, Tm(v) = 0 for all v ∈ V). For instance, the differentiation map Pn(F)→Pn(F) is nilpotent.

5. Let V be a vector space over a field F. We say a linear map T : V → V is nilpotent if there exists a positive integer m such that Tm-O (that is, Tm(v) 0 for all v V. For instance, the differentiation map Pn (F)Pn(F) is nilpotent. (a) Let V be finite-dimensional and T : V V a linear map such that for every v є V, there exists an integer k 2 1 (possibly depending on v) such that T(v) 0. Show that T is nilpotent. (b) Let dim(V-nand T : V V bea nilpotent linear map. Show that if λ is an eigenvalue of T, then λ 0. conclude that if the characteristic polynomial pr(t) of T splits over F, then pr(t) = (-1)nt. Remark: The extra hypothesis here that pr(t) splits over Fis actually not necessary, as it is automatically satisfied for a nilpotent map. See the practice problems

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