EXAM IIMath 51, Spring 2001.You have 2 hours.No notes, no books.YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONINGTO RECEIVE CREDITGood luck!NameID number1. (/20 points)2. (/20 points)3. (/20 points)4. (/20 points)5. (/20 points)Bonus (/10 points)Total (/100 points)“On my honor, I have neither given norreceived any aid on this examination. Ihave furthermore abided by all otheraspects of the honor code with respect tothis examination.”Signature:Circle your TA’s name:Kuan Ju Liu (2 and 6)Robert Sussland (3 and 7)Hunter Tart (4 and 8)Alex Meadows (10)Dana Rowland (11)Circle your section meeting time:11:00am 1:15pm 7pm11. (a) Use determinants to find the area of the triangle inR2with vertices located at15,43,−21(b) Use the cross product to determine the area of the triangle in R3with vertices locatedat124,253,0−422(c) Noticing that for vectors−→v ,−→w, and−→x, we have(−→v ×−→w) ·−→x= detx1x2x3v1v2v3w1w2w3use the properties of the determinant to show that the cross product of two vectorsis always perpendicular to each of those two vectors.32. Let the basisBbe given by the vectors {−→v1,−→v2,−→v3}, with[−→v1]S=−2/73/76/7,[−→v2]S=6/7−2/73/7,[−→v3]S=3/76/7−2/7(Note that the vectors inBare all orthogonal, and are all unit vectors.)(a) Find the matrix C which converts from Bcoordinates to S (standard basis) coordi-nates.(b) LetTbe the linear transformation which rotates vectors inR3by an angle ofπ/6radians around−→v1, in the direction from−→v2toward−→v3. What is the matrix MforTwith respect to the basisB?4(c) LetAbe the matrix for T(the transformation from part (b)) with respect to thestandard basis S. Express A in terms of MandC. (You do not need to explicitlycomputeA.) Explain.(d) LetF be the composition transformation defined byF= R◦ T, where R is thetransformation which rotates a vector by an angle ofπ/6 radians around−→e1, in thedirection from−→e2toward−→e3. What is the matrixB for the transformationF withrespect to the standard basisS? (Express your answer in terms of Mand C.)53. (a) Compute the matrix productABwhereA=1 3 46 −2 0B=4 5 23−2 14 3 1(b) LetAbe given by the 2×3 matrix below, and let Bbe the 3× n matrix with rows−→v1,−→v2,−→v3:A=a1a2a3a4a5a6B=−→v1−→v2−→v3Write the row vectors of the product ABas linear combinations of the vectors−→v1,−→v2,−→v3.64. Let the linear transformations below have matricesA, B, L, M, with domains and rangesas described in the diagram below:R4A−→R2B−→R3L−→R1M−→R3For the following problems, you might want to consider using the Rank-Nullity Theorem:(a) Find the largest possible dimension for C(A)(b) Find the smallest possible dimension for N(BA)(c) Find the smallest possible dimension forN(MLB)(d) Suppose that N(B) has dimension 1; what are the possible dimensions of C(MLBA)?Explain.75. Prove that a linear transformationT: R2→R2with matrixA=a bc dis invertible if and only if the determinantad− bc is not equal to zero.8Bonus Question– Prove or find a counterexample to the following statement:Proposition: If an n×n matrix Ahas the property thatA2= 0n(where 0nis then ×n matrix whose entries are all zero), then the matrix A must equal
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