SECOND PRACTICE EXAM SECOND HOURLY Math 21b, Fall 2003Name:MWF10 Izzet CoskunMWF11 Oliver Knill• Start by writing your name in the above box andcheck your section in the box to the left.• Try to answer each question on the same page asthe question is asked. If needed, use the back or thenext empty page for work. If you need additionalpaper, 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 forthe grader can not be given credit.• No notes, books, calculators, computers, or otherelectronic aids can be allowed.• You have 90 minutes time to complete your work.1 202 103 104 105 106 107 108 109 10Total: 100Problem 1) (20 points) True or False? No justifications are needed.T FIf A is a non-invertible n × n matrix, then det(A) 6= det(rref(A)).T FIf the rows of a square matrix form an orthonormal basis, then the columnsmust also form an orthonormal basis.T FIf the diagonal entries of an n × n matrix A are odd integers and all theentries not lying on the diagonal are even integers, then A is invertible.T FA 2 × 2 rotation matrix A 6= I2does not have any real eigenvalues.T FIf A and B both have ~v as an eigenvector, then ~v is an eigenvector of AB.T FIf A and B both have λ as an eigenvalue, then λ is an eigenvalue of AB.T FSimilar matrices have the same eigenvectors.T FIf a 3 × 3 matrix A has 3 independent eigenvectors, then A is similar to adiagonal matrix.T FIf a square matrix A has non-trivial kernel, then 0 is an eigenvalue of A.T FIf the rank of an n × n matrix A is less than n, then 0 is an eigenvalue of A.T FTwo diagonalizable matrices whose eigenvalues are equal must be similar.T FA square matrix A is diagonalizable if and only is A2is diagonalizable.T FIf a square matrix A is diagonalizable, then (AT)2is diagonalizable.T FIf a square matrix A has k distinct eigenvalues, then rank(A) ≥ (k − 1).T FThere exist matrices A with k distinct eigenvalues whose rank is strictlyless than k.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.T FIf a 3 × 3 matrix A satisfies A2= I3and A is diagonalizable, then A mustbe similar to the identity matrix.T FA and AThave the same eigenvectors.T FIf A and B are diagonalizable, AB is also diagonalizable.T FThe least squares solution of a system A~x =~b is unique if and only ifker(A) = 0.Problem 3) (10 points)Find the volume of the three dimensional parallelepiped in four dimensions which is spannedby the vectors ~u =1001, ~v =1111, ~w =1101.Problem 4) (10 points)Assume 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 is even?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)Find 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 P0of the orthogonal projection on to
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