C t 1 1 1 1 1 1 1 1 t 1 1 1 1 1 1 1 1 hence the

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(c)
T10=11100110=110T01=11100101=101Hence, the matrixAwhich representsTwith respect to the standard bases is111001.T10=11100110=110= 0100+ 1110+ 0111T11=11100111=211= 1100+ 0110+ 1111Hence, the matrixAwhich representsTwith respect to the given bases is011001.10
(d) We have to solveATAˆx=ATb:
ATA=110101111001=2112and,ATb=110101111=22.Since,212122R1R1-2R2,R2R2+2/3R1-----------------→0-3-21023we obtain,ˆx=2323.11
MULTIPLE CHOICE(? questions, 2 points each)Instructions for multiple choice questionsNo reason needs to be given. There is always exactly one correct answer.Enter your answer on the scantron sheet that is included with your exam.In addition, on your exam paper, circle the choices you made on the scantron sheet.Use anumber 2 pencilto shade the bubbles completely and darkly.DoNOTcross out your mistakes, but rather erase them thoroughly before enteringanother answer.Before beginning, please code in your name, UIN, and netid in the appropriateplaces. In the‘Section’ field on the scantron, please enter000 if Armin Straub is your instructor,001 if Philipp Hieronymi is your instructor.The actual exam will have multiple choice questions here.The midterm exams as well as the practice exams have plenty of problems that you can (andshould) look at again. Below are the short problems and multiple choice questions from theconflict exam of our midterms.Shorts 1.LetA=100110. ComputeATA.Solution.ATA=101010100110=2001Shorts 2.LetAbe a matrix such that, for everyxyzinR3,Axyz=-zx+y2x+z.Then, what isA?Solution.00-1110201Shorts 3.LetCbe a 3×3 matrix such thatChas three pivot columns, and letdbe a vectorinR3.Is it true that, if the equationCx=dhas a solution, then it has infinitely manysolutions?12
Shorts 4.LetA=a-1aaa-1.For which choice(s) ofais the matrixAnotinvertible?
det(A) = (a-1)2-a2=-2a+ 1 = 0a=12Ais invertible if and only if det(A)6= 0. Hence,Ais not invertible if and only ifa=12.Shorts 5.Write down a 3×3-matrix that is not the zero matrix (i.e. the matrix whose entriesare all zero) and is not invertible.
000111222Shorts 6.LetW1be the set of all 2×2-matricesAsuch thatAis invertible, and letW2bethe set of all 2×2-matricesAsuch thatAT=-A.Are these sets subspaces of the vectorspace of all 2×2-matrices?

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