mat67-2011-hw5solutions

mat67-2011-hw5solutions - Solutions to HW #5 Math 67 UC...

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Solutions to HW #5 Math 67 UC Davis, Fall 2011 1. Compute the coordinate vector [ v ] B , where: (a) v = (1 , 0 , 1), B = { (1 , 0 , 0) , (1 , 1 , 0) , (1 , 1 , 1) } . Solution. [ v ] B = 1 - 1 1 , since v = 1 · (1 , 0 , 0) + ( - 1) · (1 , 1 , 0) + 1 · (1 , 1 , 1). (b) v = (1 , 0 , 1), B = { (1 , 0 , 1) , (1 , 0 , 0) , (0 , 1 , 0) } . Solution. [ v ] B = 1 0 0 (c) v = z 3 - 2 z , B = { z + 1 ,z - 1 ,z 2 ,z 3 } in the space P 3 of polynomials of degree 3. Solution. [ v ] B = - 1 - 1 0 1 2. Compute the representation matrix M ( T ) B C , where: (a) T ( x,y ) = ( x + 10 y, - x ), B = C = { (1 , 0) , (0 , 1) } . Solution. M ( T ) B C = ± 1 10 - 1 0 ² (b) T ( x,y,z ) = ( z,y, 3 x ), B = { (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) } , C = { (1 , 0 , 1) , (0 , 1 , 0) , ( - 1 , 0 , 1) } Solution. M ( T ) B C = 3 2 0 1 2 0 1 0 3 2 0 - 1 2 (c) T ( x,y,z ) = ( z,y, 3 x ), B = { (1 , 0 , 1) , (0 , 1 , 0) , ( - 1 , 0 , 1) } , C = { (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1)
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mat67-2011-hw5solutions - Solutions to HW #5 Math 67 UC...

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