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Question: let f be a field andv a finitedimensional vector space...

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Let F be a field andV a finite-dimensional vector space over F. Let V denote the dual space of V (i.e. V v is the set of all linear maps V → F, with addition and scalar multiplication defined as follows: given f, g E VV and cE F, the maps f+g:VF and cf: V Fare given by (f +g(v)-f(v) g(v) and (cf)(v)c.f(. Let T: V-V be a linear operator. Then given any f E VV, being a composition of linear transformations, f。T : V → F is also linear. Let Tt (called the transpose or dual of T) be the map νν → Vv defined by Tt (f) =f-T. You can check that T is indeed linear (but you dont have to include the argument in your solution). Show that the characteristic polynomials of T and T are equal. (Suggestion: Let B be a basis of V. Let y be the basis of V v dual to β. Try to relate [T]P and [T, .)
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