HW3Soln - = 0. 2 8. Problem 2.2.42 If Ax 1 = b and Ax 2 = b...

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ECE 130C HW-3 Solution April 30, 2008 1. Problem 2.1.20 (a) True (b) False(consider A + A T ) (c) True: Ax = 0 andBx = 0 = ( cA + dB ) x = 0 2. Problem 2.1.26 C ( AB ) C ( A ) If B = 0, then C ( AB ) C ( A ) 3. Problem 2.1.28 (a) False (b) True (c) True (d) False take A = I 4. Problem 2.2.2 Echelon form: U = 1 2 0 1 0 1 1 0 0 0 0 0 u and v are the pivot variables; w and y are free variables. 1
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The complete solution is x = 2 w - y - w w y = w 2 - 1 1 0 + y - 1 0 0 1 Rank ( A ) = 2 EchelonForm,U = 1 2 3 0 - 3 - 6 0 0 0 w is free, all solutions are ( w, - 2 w,w ). 5. Problem 2.2.8 No condition on b 1 ,b 2 ( - 2 , 1 , 0 , 0) , (0 , 0 , 1 , 0) are basis for N ( A ). The complete solution is x = 7 b 1 - 3 b 2 0 0 b 2 - 2 b 1 + v - 2 1 0 0 + w 0 0 1 0 N ( A T ) = (0 , 0). 6. Problem 2.2.10 Rank is 2. A can be ± 1 0 - 1 0 1 - 3 ² with b = ± 1 2 ² . 7. Problem 2.2.20 S = ± 1 3 1 4 ² and S = ³ 1 ´ and S = ± 1 0 0 1 ² . itemProblem 2.2.36 (a) Every b is in the column space: independent rows . (b) Need b 3 = 2 b 2 for ( b 1 ,b 2 ,b 3 ) to be in the column space. Row 3 - 2*Row 2
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Unformatted text preview: = 0. 2 8. Problem 2.2.42 If Ax 1 = b and Ax 2 = b then we can add c ( x 1-x 2 ) to any solution of Ax = B . But there will be no solution to Ax = B if B is not in the column space. 9. Problem 2.2.50 A = 1 0 3 0 . 10. Problem 3 (not in text) A m * n = RX where R is an m * r matrix with the columns as the basic columns of A and R is an r * n matrix with the r rows corresponding to the non-zeros rows from the echelon form of A . Note that a direct LU factorizarion may not always be possible for all A and a permutation may be needed. 1 3 3 2 2 6 9 7-1-3 3 4 = 1 3 2 9-1 3 1 3 0-1 0 0 1 1 3...
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This note was uploaded on 08/08/2010 for the course ECE PROF. VOLK taught by Professor Volkanrodoplu during the Summer '10 term at UCSB.

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HW3Soln - = 0. 2 8. Problem 2.2.42 If Ax 1 = b and Ax 2 = b...

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