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HARVARD MATH 21B - Second Mid-term

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2nd_practice12nd_practice1_tempPRACTICE EXAMINATION ONE FOR SECOND MID-TERM November 28, 2007 Math 21b, Fall 2007 MWF10 Evan Bullock MWF11 Leila Khatami MWF12 Yum-Tong Siu 1 20 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 Total 100 • Start by writing your name in the above box and check your section in the box to the left. • Try to answer each question on the same page as the question is asked. If needed, use the back or the next empty page for work. If you need additional paper, write your name on it. • Do not detach pages from this exam packet or un-staple the packet. • Please write neatly. Answers which are illegible for the grader can not be given credit. • No notes, books, calculators, computers, or other electronic aids can be allowed. • You have 90 minutes time to complete your work.Problem 1) (20 points) True or False? No justifications are needed.1)T FIf x∗is the least squares solution of Ax = b, then||b||2= ||Ax∗||2+ ||b − Ax∗||2.2)T FSimilar matrices have the same determinant.3)T FIf λ is an eigenvalue of A, then λ3is an eigenvalue of A3.4)T FA shear in the plane is not diagonalizable.5)T FIf A is invertible, then A and A−1have the same eigenvectors.6)T FIf A is a 3 × 3 matrix for which every entry is 1, then det(A) = 1.7)T FIf ~v is an eigenvector of A and of B and A is invertible, then ~v is aneigenvector of 3A−1+ 2B.8)T Fdet(−A) = det(A) for every 5 × 5 matrix A.9)T FIf ~v is an eigenvector of A and an eigenvector of B and A is invertible, then~v is an eigenvector of A−3B2.10)T FIf a 11 × 11 matrix has the eigenvalues 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, then A isdiagonalizable.11)T FFor any n × n matrix, the matrix A has the same eigenvectors as AT.12)T FIf ~v =abcis a vector of length 1, then ~v~vTis a diagonalizable 3 × 3matrix.13)T FThe span of m orthonormal vectors is m-dimensional.14)T FA square matrix A can always be expressed as the sum of a symmetricmatrix a nd a skew-symmetric matrix as follows A =12(A+AT)+12(A−AT).15)T FThere exists an invertible n × n matrix A which satisfies ATA = A2, but Ais not symmetric.16)T FIf two n × n matrices A and B commute, then (AT)2commutes with (B3)T.17)T F−AATis skew-symmetric for every n × n matrix A.18)T FA matrix which is obtained f r om the identity matrix by an arbitrary numberof switching of rows or columns is an orthogonal matrix.19)T FThere exists a real 3 × 3 matrix A which satisfies A4= −I3.20)T FGiven 5 data points (x1, y1), ..., (x5, y5), then a best fit with a polynomiala + bt + ct2+ dt3+ et4+ ft5is possible in a unique way.2Problem 2) (10 points)Check the boxes of all matrices which have zero determinants. You don’t have to give justifi-cations.a)A =1 0 1 00 0 1 10 1 0 00 0 1 0b)A =1 2 3 45 6 7 89 10 11 1213 14 15 16c)A =1 1 1 10 1 1 00 1 1 01 1 1 1d)A =0 0 0 14 0 0 00 3 1 02 0 1 1e)A =101001 1 11 101001 11 1 1010001 0 1 10100f)A =1 1 0 02 3 0 00 0 1 10 0 1 1g)A =11 10 8 59 7 4 06 3 0 02 0 0 0h)A =1 1 1 11 1 −1 −11 −1 1 −11 −1 −1 1i)A =1 1 7 01 6 0 05 0 0 00 0 0 10j)A =2 3 4 52 3 4 52 3 4 52 3 4 5Problem 3) (10 points)a) Find all (possibly complex) eigenvalues and eigenvectors of the matrix Q =0 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 11 0 0 0 0 0b) Verify that QThas the same eigenvectors as Q.c) Find a diagonal matrix B which is similar to the symmetric matrixA = Q + QT=0 1 0 0 0 11 0 1 0 0 00 1 0 1 0 00 0 1 0 1 00 0 0 1 0 11 0 0 0 1 0.3State algebraic and geometric multiplicities of the eigenvalues.Problem 4) (10 points)a) Let A be a n × n matrix such that A2= 2A − I. What are the possible eigenvalues of A?b) Let A be a real n × n matrix such that A4= −In. Show that n must be even.Problem 5) (10 points)Find the function f (t) = a + bt which best fits the data(x1, y1) = (−1, 1)(x2, y2) = (0, 2)(x3, y3) = (1, 2)(x4, y4) = (3, 1)(x5, y5) = (3, 0)Problem 6) (10 points)Find the determinant of the matrix1 2 3 4 5 6 7 8−1 0 3 4 5 6 7 8−1 −2 0 4 5 6 7 8−1 −2 −3 0 5 6 7 8−1 −2 −3 −4 0 6 7 8−1 −2 −3 −4 −5 0 7 8−1 −2 −3 −4 −5 −6 0 8−1 −2 −3 −4 −5 −6 −7 0.Show your work carefully.4Problem 7) (10 points)A discrete dynamical system is given byT ("xy#) ="2x − 2y−x + 3y#= A"xy#.Find a closed formula for T100("12#).Problem 8) (10 points)a) Show that for an arbitrary matrix A for which ATA is invertible, the least squares solutionof A~x =~b simplifies to~x = R−1QT~b ,if A = QR is the QR decomposition of A.b) Find the least square solution in the case A =1 10 10 1and~b =121using this f ormula.Problem 9) (10 points)a) (7 points) Find the QR decomposition ofA ="2 12 3#then for m the new matrix T (A) = RQ.b) (3 points) Verify that for any invertible n × n matrix A = QR, the matrix T (A) = RQ hasthe same eigenvalues as A.c) Find the least square solution for the system A~x =~b g iven by the equationsx + y = 4y = 2x = −1


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HARVARD MATH 21B - Second Mid-term

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