2nd_practice22nd_practice2_tempPRACTICE EXAMINATION TWO 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 A is a non-invertible n × n matrix, then det(A) 6= det(rref(A)).2)T FIf the rows of a square matrix form an orthonormal basis, then the columnsmust also form an orthonormal basis.3)T FA 3 × 3 matrix A for which the sum of the first two columns is the thirdcolumn has zero determinant.4)T FA 2 × 2 rotation matrix A 6= I2does not have any real eigenvalues.5)T FIf A and B both have ~v as an eigenvector, then ~v is an eigenvector of AB.6)T FIf A and B both have λ as an eigenvalue, then λ is an eigenvalue of AB.7)T FSimilar matrices have the same eigenvect ors.8)T FIf a 3 × 3 matrix A has 3 independent eigenvectors, then A is similar to adiagonal matrix.9)T FIf a square matrix A has non-trivial ker nel, then 0 is an eigenvalue of A.10)T FIf the rank of an n × n matrix A is less than n, then 0 is an eigenvalue of A.11)T FTwo diagonalizable matrices whose eigenvalues are equal must be similar.12)T FA square matrix A is diagonalizable if and only is A2is diagonalizable.13)T FIf a square matrix A is diagonalizable, then (AT)2is diagonalizable.14)T FThe matrices"3 20 3#and"3 00 3#are similar.15)T FThere exist matrices A with k distinct eigenvalues whose rank is strictlyless than k.16)T FIf A is an n × n matrix which satisfies Ak= 0 for some positive integer k,then all the eigenvalues of A are 0.17)T FIf a 3 × 3 matrix A satisfies A2= I3and A is diagonalizable, then A mustbe similar to the identity matrix.18)T FA and AThave the same eigenvectors.19)T FThe least squares solution of a system A~x =~b is unique if and only ifker(A) = 0.20)T FThe matrix1 1000 1 11000 1 1 11 1 1 10001 1 1000 1is invertible.2Problem 2) (10 points)Match the following matrices with the correct label. No justifications are needed. Fill ina),b),c),d),e) into the boxes.A)2 3 43 5 74 7 9B)0 3 4−3 0 7−4 −7 0C)1 0 00 0 −10 1 0D)1 0 00 1 00 0 0E)1 1 00 1 10 0 1a) (2 points) skewsymmetric matrixb) (2 points) nondiagonalizable matrixc) (2 points) orthogonal projectiond) (2 points) symmetric matrixe) (2 points) orthogonal matrixProblem 3) (10 points)Find a basis for the subspace V of R4given by the equation x + 2y + 3z + 4w = 0. Find thematrix which gives the orthog onal projection onto this subspace.Hint: The problem can be done in different ways. Choosing an orthonormal basis in Vsimplifies some computations.Problem 4) (10 points)3Assume that A is a skew-symmetric matrix, that is, it is a n×n matrix which satisfies AT= −A.a) Find det(A) if n is odd.b) What possible values can det(A) have if n = 2?c) Verify that if λ is an eigenvalue of A, then −λ is also an eigenvalue of A.Problem 5) (10 points)The recursionun+1= un− un−1+ un−2is equivalent to the discrete dynamical systemun+1unun−1=1 −1 11 0 00 1 0unun−1un−2= Aunun−1un−2.a) Find the (real or complex) eigenvalues of A.b) Is there a vector ~v such that ||An~v|| → ∞?c) Can you find any positive integer k such that Ak= I3?Problem 6) (10 points)Let A be the matrixA =0 1 0 00 0 1 00 0 0 11 0 0 0.a) Find det(A).b) Find all eigenvalues whether real or complex of A and state their algebraic multiplicities.c) For each real eigenvalue λ of A find the eigenspace and the geometric multiplicity.Problem 7) (10 points)4Find S and a diagonal matrix B such that S−1AS = B, whereA =1 0 01 2 01 2 3.Problem 8) (10 points)Find the function of the formf(t) = a sin(t) + b cos ( t) + cwhich best fits the data points (0, 0), (π, 1), (π/2, 2), (−π, 3).Problem 9) (10 points)Let V be the image of the matrixA =1 01 11 01 1.a) Find the matrix P of the orthogonal projection onto V .b) Find the matrix P′of the orthogo na l projection on to
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