assign5_soln

assign5_soln - Math 136 Assignment 5 Solutions 1 Let A = 1...

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Unformatted text preview: Math 136 Assignment 5 Solutions 1. Let A = 1 2- 3 4 5 2 , B = 3- 1 1 2 , C = - 1- 1 1 2- 2 2 . Determine the following a) 2 A- 3 B Solution: 2 A- 3 B = 2 4- 6 8 10 4 - 9- 3 3 6 = - 7 7- 6 5 10- 2 b) A ( B T + C T ) Solution: A ( B T + C T ) = 1 2- 3 4 5 2 3 0 0- 1 1 2 +- 1 1- 2- 1 2 2 = 1 2- 3 4 5 2 2 1- 2- 2 3 4 = - 2 7 6- 14 9 22 6 11- 2 c) BA T + CA T Solution: BA T + CA T = ( AB T ) T +( AC T ) T = ( AB T + AC T ) T = ( A [ B T + C T ]) T = - 2- 14 6 7 9 11 6 22- 2 . 2. Prove that if ~x,~ y ∈ M (2 , 3) and s ∈ R are scalars, then s ( ~x + ~ y ) = s~x + s~ y . Solution: Let ~x = x 11 x 12 x 13 x 21 x 22 x 23 and ~ y = y 11 y 12 y 13 y 21 y 22 y 23 . Then s ( ~x + ~ y ) = s x 11 + y 11 x 12 + y 12 x 13 + y 13 x 21 + y 21 x 22 + y 22 x 23 + y 23 = sx 11 + sy 11 sx 12 + sy 12 sx 13 + sy 13 sx 21 + sy 21 sx 22 + sy 22 sx 23 + sy 23 = s x 11 x 12 x 13 x 21 x 22 x 23 + s y 11 y 12 y 13 y 21 y 22 y 23 = s~x + s~ y 1 2 3. For each of the following mappings: (i) Prove that it is linear. (ii) Find the standard matrix of the mapping (iii) Find a spanning set for the nullspace. (iv) Find a spanning set for the range. a) f ( x 1 , x 2 , x 3 ) = ( x 1 , x 2 , x 1 + x 2 + x 3 ). Solution: (i) Let ~x,~ y ∈ R 3 and s ∈ R , then f ( s~x + ~ y ) = f sx 1 + y 1 sx 2 + y 2 sx 3 + y 3 = sx 1 + y 1 sx 2 + y 2 sx 1 + y 1 + sx 2 +...
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assign5_soln - Math 136 Assignment 5 Solutions 1 Let A = 1...

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