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midterm_B - S a subspace of R 4 Show work to justify your...

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310B Linear Algebra and Applications Midterm Examination - Oct 16, 2007 Answer all the questions. Duration: 1 hour Name: Student ID: Class: 1. Let A = 2 - 1 2 35 - 2 - 4 6 - 4 1 , b = 1 4 3 . (a) Find an LU factorization of A . (b) Use 1(a) to solve A x = b . 2. Find a condition or conditions on α so that the system x 1 + x 2 + αx 3 = α 2 x 1 + αx 2 + x 3 = α αx 1 + x 2 + x 3 = 1 has (a) a unique solution; (b) no solution; (c) an infinite number of solutions. 3. Let v 1 = 1 0 - 1 , v 2 = 2 1 3 , v 3 = 0 - 1 1 , v 4 = 1 1 1 . (a) Is the set { v 1 , v 2 , v 3 , v 4 } linearly independent or linearly dependent? Show work to justify your answer. (b) Find one vector v i that is a linear combination of the other v j , if this is possible. Give the explicit linear combination. (c) Does the vector k = [1 1 - 1] T lie in W = span { v 2 , v 3 , v 4 } ? Show work to justify your answer.
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2 4. Let S = a c b d fl fl fl fl fl fl fl fl a + 2 b - 3 c + d = 0 (a) Is
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Unformatted text preview: S a subspace of R 4 ? Show work to justify your answer. (b) Find a list of vectors which span S . 5. Answer each question separately. Show work to justify your answers . (a) Suppose the matrix A =   a b c p q r x y z   has cofactor matrix   D E F G H N J K M   . What is aE + pH + xK ? (b) Try to find two nonsingular 2 × 2 matrices A and B with ( A + B ) 2 = A 2 + B 2 . (c) If A is an n × n skew symmetric matrix and n is odd, show that A must be singular. (d) Let A =   λ 1 0 λ 1 0 0 λ   . Find A k for k ≥ 2. 6. Find the determinant of A =     1 + x 1 x 2 x 3 x 4 x 1 1 + x 2 x 3 x 4 x 1 x 2 1 + x 3 x 4 x 1 x 2 x 3 1 + x 4     and express your final answer in the most compact and simplified form....
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