A0 - 1 1 c C = 2 1-3 5 1-2-1 3-4 3 1-3 3-1 4 2-6 12 4-3-1...

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Math 235 Assignment 0 Due: Not To Be Submitted 1. Determine proj ~v ~x and perp ~v ~x where a) ~v = (2 , 3 , - 2) and ~x = (4 , - 1 , 3). b) ~v = ( - 1 , 2 , 1 , - 3) and ~x = (2 , - 1 , 2 , 1). 2. Prove algebraically that proj ~v ( ~x ) and perp ~v ~x are orthogonal. 3. Solve the system z 1 - z 2 + iz 3 = 2 i (1 + i ) z 1 - iz 2 + iz 3 = - 2 + i (1 - i ) z 1 + ( - 1 + 2 i ) z 2 + (1 + 2 i ) z 3 = 3 + 2 i 4. Prove that proj (2 , 2 , - 1) is linear and find its standard matrix. 5. Let S = { ( a, b ) R 2 | b > 0 } and define addition by ( a, b ) + ( c, d ) = ( ad + bc, bd ) and define scalar multiplication by k ( a, b ) = ( kab k - 1 , b k ). Prove that S is a vector space over R . 6. Prove each of the following are subspaces of the given vector space and find a basis for each. a) S = ±² a b c d ³ ´ ´ ´ ´ a + b + c + d = 0 µ of M (2 , 2). b) T = ± p ( x ) P 2 ´ ´ ´ ´ p (2) = 0 µ of P 2 . 7. Find the inverse of the following matrices. a) A = ² 2 5 - 3 2 ³ . b) B = 2 - 1 3 1 2 2 1 0 1 .
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2 8. Find the determinant of the following matrices. a) A = ± 2 5 - 3 2 ² . b) B = 2 - 1 3 1 2 2
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Unformatted text preview: 1 1 . c) C = 2 1-3 5 1-2-1 3-4 3 1-3 3-1 4 2-6 12 4-3-1 3-3 9. Diagonalize the following matrices a) A = 1 6 3-2 0 3 6 1 . b) B = -7 2 12-3 0 6-3 1 5 . 10. Prove the following. a) Let A be an n × n matrix. If rank A < n , then 0 is an eigenvalue of A . b) If A is an n × n matrix with n distinct real eigenvalues λ 1 , . . . , λ n , then det A = λ 1 · · · λ n . c) For any vectors ~v 1 ,~v 2 in a vector space V we have span { ~v 1 ,~v 2 } = span { k~v 1 , a~v 1 + b~v 2 } , for any constants a, b, k , with b and k non-zero....
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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A0 - 1 1 c C = 2 1-3 5 1-2-1 3-4 3 1-3 3-1 4 2-6 12 4-3-1...

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