Linear Algebra (Math 251)

Ministry of Higher Education Kingdom of Saudi Arabia CSTS SEU, KSA Linear Algebra (Math 251) Level IV, Assignment 3 (2015-16) 1. State whether the following statements are true or false: (a) The product of eigen values of a matrix is same as its determinants. (a) (b) The eigen values of the matrix A =   2 0 0 6 −1 0 17 3 4   are 2, 4 and 0. (b) (c) The inner product of two vectors cannot be a negative real number (c) (d) If v = (3, 4) then kvk = 5. (d) (e) In an inner product space (V, <, >) if x and y are unit vectors orthogonal to each other then kx + yk = 2. (e) (f) If u = (4, 3, 1, −2) and v = (−2, 1, 2, 3) then < u, v >= −9. (f) (g) The matrix A =   7 1 − i 8 1 − i 5 −1 − 6i 8 6i − 1 −1   is Hermitian. (g) (h) A square matrix A is orthogonal, if A−1 = AT . (h) (i) The matrix A is unitary if A∗ = A. (i) Page 1 of 3 Please go on to the next page. . . Math 150 Department of Mathematics 2. Select one of the alternatives from the following questions as your answer. (a) Which of the following sets of vectors are orthogonal with respect to the Euclidean inner product on R2 : A. (0,6), (7,0) B. (3,4),(2,6) C. (6,9),(5,2) D. (0,4), (0,6) (b) If kuk = √ 30, kvk = √ 18 and < u, v >= −9, then cos θ = A. −2 3 √ 15 B. −3 2 √ 15 C. −2 3 √ 60 D. None (c) The values of k for which u = (k, −4, 8) and v = (k, k, −4) are orthogonal in R 3 Euclidean Inner Product Space are A. 8, -4 B. 4, -8 C. -4, -8 D. 4, 8 (d) The eigen values of a Hermitian matrix are A. complex only B. complex and real both C. always zero D. always real (e) If 0 is an eigen value of a square matrix A then A is A. an Identity matrix. B. invertible. C. not invertible. D. None (f) If square matrix A is such that AA∗ = I, then A is A. Hermitian B. Unitary C. skew-symmetric D. None Page 2 of 3 Please go on to the next page. . . Math 150 Department of Mathematics (g) The matrix A =   1 9 8 9 − 4 9 4 9 − 4 9 − 7 9 8 9 1 9 4 9   is A. Hermitian B. Unitary C. skew-symmetric D. Orthogonal 3. Compute < U, V > using the inner product on M2×2, where U =  9 −8 9 18  and V =  −1 9 1 1  . 4. Let R3 have the Euclidean inner product. Find the cosine of the angle θ between u = (−1, 6, 2) and v = (4, 3, −5). 5. Find all the least squares solution of the linear system x1 − x2 = 2 2x1 + 3x2 = −1 4x1 + 5x2 = 2. 6. Let R3 have the Euclidean inner product. For which values of k are u and v orthogonal? (a) u = (2, 1, 3), v = (1, 7, k). (b) u = (k, k, 1), v = (k, 5, 6). 7. If P2 have the usual inner product on polynomials and p = 1 − 2x + 3x 2 , q = 3 + x 2 are the polynomials. Then find (a) kpk (b) kqk (c) < p, q > 8. Show that A =   1 1 2 + 3i i −3 1 2 − 3i 1 2   is Hermitian. Page 3 of 3 End of Assignment.