# A0 - a) A = 2 5-3 2 . b) B = 2-1 3 1 2 2 1 1 . c) C = 2 1-3...

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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 each of the following mappings are linear. a) proj (2 , 2 , - 1) . b) L ( ~x ) = A~x , where A is an m × n 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, ﬁnd a basis for each, and determine the dimension. 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 . c) S 1 = x 1 x 2 x 3 R 3 | x 1 + x 2 = 0 ,x 3 = - x 2 of R 3 .

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2 7. Find the inverse of the following matrices. a) A = ± 2 5 - 3 2 ² . b) B = 2 - 1 3 1 2 2 1 0 1 . 8. Find the determinant of the following matrices.
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Unformatted text preview: a) A = 2 5-3 2 . b) B = 2-1 3 1 2 2 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 1 ,..., n , then det A = 1 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 11/28/2011 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

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A0 - a) A = 2 5-3 2 . b) B = 2-1 3 1 2 2 1 1 . c) C = 2 1-3...

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