MH1201 - Solution Set 9 - MH1201 - SOLUTIONS TO PROBLEM SET...

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MH1201 - SOLUTIONS TO PROBLEM SET 9Problem 1.(1) We haveT(β1) =T12=-2-3= (-1)β2=0β1+ (-1)β2T(β2) =T23=24= 2β1=2β1+ 0β2,and hence[T]β="02-10#.(2) It is clear, by 1), that the basisβdoes not consist of eigenvectors ofT.Problem 2.It’s similar to Problem 1.(1) We haveT(β1) =T(1-x+x3) =-1 +x-x3=-β1=(-1)β1+ 0β2+ 0β3+ 0β4T(β21) =T(1 +x2) =-x-x2+x3=β1-β2=1β1+ (-1)β2+ 0β3+ 0β4T(β3) =T(1) =x2=β2-β3=0β1+ 1β2+ (-1)β3+ 0β4T(β4) =T(x+x2) =-x-x2=-β4=0β1+ 0β2+ 0β3+ (-1)β4,and hence[T]β=-11000-11000-10000-1.(2) It is clear, by 1), that the basisβdoes not consist of eigenvectors ofT.Problem 3.It’s similar to Problem 4, Tutorial # 8 last week. We haveA="1232#.
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