# MH1201 - Problem Set 9 (updated 22 Mar) - MH1201 - PROBLEM...

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MH1201 - PROBLEM SET 9Date: 26–28 Mar 2014Topic: Eigenvalues, eigenvectors (continued)Problem 1.Consider the vector spaceR2with the order basisβ=("12#,"23#)and a linear operatorT:R2R2defined byT"ab#="10a-6b17a-10b#.(1) Compute [T]β.(2) Determine whetherβis a basis consisting of eigenvectors ofT.Problem 2.Consider the vector spaceP3(R) with the order basisβ=n1-x+x3,1 +x2,1, x+x2oand a linear operatorT:P3(R)P3(R) defined byT(a+bx+cx2+dx3) =-d+ (-c+d)x+ (a+b-2c)x2+ (-b+c-2d)x3.(1) Compute [T]β.(2) Determine whetherβis a basis consisting of eigenvectors ofT.Problem 3.Given a matrixAM2×2(R), whereA="1232#.(1) Determine all the eigenvalues ofA.(2) For each eigenvalueλofA, find the set of eigenvectors corresponding toλ.(3) If possible, find a basis forR2consisting of eigenvectors ofA.(4) If successful in finding such a basis, determine an invertible matrixQand a diagonalmatrixDsuch thatQ-1AQ=D.Problem 4.For a linear operatorT:R2R2defined byT"ab#="-2a+ 3b-10a+ 9b#find the eigenvalues ofTand an ordered basisβforR2so that [T]βis a diagonal matrix.Problem 5.For a linear operator
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