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# HW02 - 2 = a b 2 c,a-b Let β and γ be the standard...

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EE 203000 Linear Algebra Homework #2 (Due October 28, 2009 BEFORE class) Note: Detailed derivations are required to obtain a full score for each problem. (Total 100%) 1. (4%+4%+6%) Problem 31 (a),(b),(d) of Section 1.3. 2. (5%+5%) Problem 31 of Section 1.6. 3. (6%+6%) Let V and W be vector spaces and let T : V W . Prove the following properties regarding the linear transformation. (i) If T is linear, then T (0 ) = 0 . (ii) T is linear if and only if T ( cx + y ) = cT ( x ) + T ( y ), for all x ,y V and c F . 4. (6%+6%) Problem 25 (a), (c) of Section 2.1. 5. (6%) Problem 8 of Section 2.2. 6. (4%+4%) Let A = p 1 3 2 - 1 P , B = p 1 0 - 3 4 1 2 P , C = p 1 0 4 - 1 - 2 0 P , and D = 2 - 1 3 . Compute A (2 B + C ) and A ( BD ). (Show your derivations step-by-step.) 7. (4%+4%) Let g ( x ) = 5 + 2 x . Let T : P 2 ( R ) P 2 ( R ) and U : P 2 ( R ) R 3 be the linear transformations de±ned by T ( f ( x )) = f ( x ) g ( x ) + 2 f ( x ) and U ( a + bx + cx

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Unformatted text preview: 2 ) = ( a + b, 2 c,a-b ) . Let β and γ be the standard ordered bases for P 2 ( R ) and R 3 , respectively. (a) Compute [ U ] γ β , [ T ] β , and [ UT ] γ β directly. Verify your results with Theorem 2.11. (b) Let h ( x ) = 3-2 x + x 2 . Compute [ h ( x )] β and [ U ( h ( x ))] γ . Verify your results with Theorem 2.14. 1 8. (4%+4%) Problem 13 of Section 2.3. 9. (4%+6%) (a) Show that T : R 3 → R 3 deFned by T ( a 1 ,a 2 ,a 3 ) = (3 a 1-2 a 3 ,a 2 , 3 a 1 +4 a 2 ) is invertible. (b) Let A and B be n × n invertible matrices. Prove that AB is invertible and ( AB ) − 1 = B − 1 A − 1 . 10. (6%+6%) Problem 17 of Section 2.4. [Moved to HW#3] 2...
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HW02 - 2 = a b 2 c,a-b Let β and γ be the standard...

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