# Pts consider the linear transformation t r 3 p 3 r

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Problem 1 (20 pts).Consider the linear transformationT:R3P3(R) given byTabc=a+bx+ (a+b)x3.(a) Find the matrix [T]γβofTrelative to the standard basesβandγofR3andP3(R).(b) Explain whyTis neither 1-1 nor onto.3
Problem 2 (30 pts).LetVbe the set of matrices defined as follows:V=a-bbaa, bR.(a) Show thatVis a vector space overR.(b) Give a basis forVand find dimV.4
(c) ConsiderC, the set of complex numbers, as a vector space overR.Give a basis forCand find its dimension.(d) Letf:VCbe a function defined byfa-bba=a+ibfor alla, bR.Prove thatfis an isomorphism of vectors spaces overR.5
Problem 3 (20 pts).LetVbe a vector space and letL(V, V) be the vector space of all lineartransformations fromVtoV.(a) State what it means for several linear transformationsT1, T2, . . . , TkfromVtoVto belinearly independent inL(V, V).(b) LetV=P5(R). LetTandUbe linear transformations fromVtoVdefined byT(f(x)) =f(2) +f0(x) andU(f(x)) =f00(x) for all polynomialsf(x)V.Prove thatTandUare linearly independent inL(V, V).6
Problem 4 (10 pts).LetVbe a vector space of dimension 2017. LetT:VVbe a lineartransformation and let~vbe some vector inV. Prove that for a large enough positive integermthevectors~v, T(~v), T2(~v), T3(~v), . . . , Tm(~v)will be linearlydependentinV.7