MH1201 - Problem Set 8 - MH1201 PROBLEM SET 8 Date 1921 Mar 2014 Topics Eigenvalues eigenvectors Problem 1 Is it true that every linear operator on an

# MH1201 - Problem Set 8 - MH1201 PROBLEM SET 8 Date 1921 Mar...

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MH1201 - PROBLEM SET 8 Date: 19–21 Mar 2014 Topics: Eigenvalues, eigenvectors Problem 1.Is it true that every linear operator on ann-dimensional vector space hasndistinct eigen-values? Justify your answer.Problem 2.Is it true that if a real matrix has one eigenvector, then it has an infinite number of eigen-vectors? Justify your answer.Problem 3.Given a linear operatorT∈ L(V) on a vector spaceVand a basisβfor this vector space,calculate the matrix representation ofTwith respect to the basisβ, and determine whetherβis a basis of eigenvectors forT.(1)V=R3,Tabc=3a+ 2b-2c-4a-3b+ 2c-c, andβ=011,1-10,102.(2)V=P1(R),T(a+bx) = (6a-6b) + (12a-11b)x, andβ={3 + 4x,2 + 3x}.Problem 4.Given a matrixAM3×3(R), whereA=0-2-3-11-1225.(1) Determine all the eigenvalues ofA.(2) For each eigenvalueλofA, find the set of eigenvectors corresponding toλ.(3) If possible, find a basis forR3consisting of eigenvectors ofA.(4) If successful in finding such a basis, determine an invertible matrix