A0_soln - Math 235 Assignment 0 Solutions 1 Determine proj...

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Unformatted text preview: Math 235 Assignment 0 Solutions 1. Determine proj ~v ~x and perp ~v ~x where a) ~v = 2 3- 2 and ~x = 4- 1 3 . Solution: proj ~x ~v = ~x · ~v k ~v k 2 ~v =- 1 17 2 3- 2 = - 2 / 17- 3 / 17 2 / 17 b) ~v = - 1 2 1- 3 and ~x = 2- 1 2 1 .. Solution: proj ~x ~v = ~x · ~v k ~v 2 k 2 ~v =- 1 15 - 1 2 1- 3 = 1 / 15- 2 / 15- 1 / 15 3 / 15 . 2. Prove algebraically that proj ~x ( ~v ) and perp ~x ~v are orthogonal. Solution: We have proj ~x ( ~v ) · perp ~x ~v = ~v · ~x k ~x k 2 ~x · ~v- ~v · ~x k ~x k 2 ~x = ~v · ~x k ~x k 2 ( ~x · ~v )- ~v · ~x k ~x k 2 2 ( ~x · ~x ) = ( ~v · ~x ) 2 k ~x k 2- ( ~v · ~x ) 2 k ~x k 4 k ~x k 2 = ( ~v · ~x ) 2 k ~x k 2- ( ~v · ~x ) 2 k ~x k 2 = 0 . Hence, they 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 2 Solution: Making an augmented matrix and row-reducing we get 1- 1 i 2 i 1 + i- i i- 2 + i 1- i- 1 + 2 i 1 + 2 i 3 + 2 i ∼ 1 0 1 + i i 0 1 1- i 0 0 x 3 does not have a leading one so let x 3 = t ∈ C . Then we have x 1 = i- (1 + i ) t x 2 =- i- t x 3 = t So the general solution is ~x = i- i + t - 1- i- 1 1 . 4. Prove each of the following mappings are linear and find the standard matrix of each. a) proj (2 , 2 ,- 1) . Solution: Let ~n = 2 2- 1 and let ~x,~ y ∈ R 3 and k ∈ R . Then by using properties of the dot product we get proj (2 , 2 ,- 1) ( k~x + ~ y ) = ( k~x + ~ y ) · ~n ) k ~n k 2 ~n = k ~x · ~n k ~n k 2 ~n + ~ y · ~n k ~n 2 k 2 ~n = k proj ~n ~x + proj ~n ~ y Hence, proj (2 , 2 ,- 1) is linear. We have proj (2 , 2 ,- 1) ~e 1 = ~e · ~n ) k ~n k 2 ~n = 2 9 2 2- 1 = 4 / 9 4 / 9- 2 / 9 proj (2 , 2 ,- 1) ~e 2 = ~e · ~n ) k ~n k 2 ~n = 2 9 2 2- 1 = 4 / 9 4 / 9- 2 / 9 proj (2 , 2 ,- 1) ~e 3 = ~e · ~n ) k ~n k 2 ~n =- 1 9 2 2- 1 = - 2 / 9- 2 / 9 1 / 9 3 Hence [proj (2 , 2 ,- 1) ] = 4 / 9 4 / 9- 2 / 9 4 / 9 4 / 9- 2 / 9- 2 / 9- 2 / 9 1 / 9 ....
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This note was uploaded on 09/19/2011 for the course MATH 235 taught by Professor Celmin during the Winter '08 term at Waterloo.

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A0_soln - Math 235 Assignment 0 Solutions 1 Determine proj...

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