Math121sol5

# Math121sol5 - Math 121 Linear Algebra and Applications Prof...

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Unformatted text preview: Math 121: Linear Algebra and Applications Prof. Lydia Bieri Solution Set 5 Posted: Fri. Nov. 15, 2007 Written by: Luca Candelori Exercise 1 (2.5/3) . (a) This matrix is simply the matrix with respect to β of the linear transformation T that maps β to β . In particular T ( x 2 ) = a 2 x 2 + a 1 x + a = ( a 2 ,a 1 ,a ) T ( x ) = b 2 x 2 + b 1 x + b = ( b 2 ,b 1 ,b ) T (1) = c 2 x 2 + c 1 x + c = ( c 2 ,c 1 ,c ) so that our matrix [ T ] β looks like: a 2 b 2 c 2 a 1 b 1 c 1 a b c (b) Similarly, the matrix is a b c a 1 b 1 c 1 a 2 b 2 c 2 Exercise 2 (2.5/6a) . To find the matrix [ L A ] β , of left multiplication by A with respect to β , we look at the action of A on β : 1 3 1 1 1 1 = 4 2 = 6 · 1 1- 2 · 1 2 and 1 3 1 1 1 2 = 7 2 = 11 · 1 1- 4 · 1 2 therefore [ L A ] β = 6 11- 2- 4 The matrix Q is just the change of basis matrix, which maps (1 , 0) to (1 , 1) and (0 , 1) to (1 , 2). It looks like Q = 1 1 1 2 1 Exercise 3 (2.6/2bcd) . (b) FALSE. This function is linear but it maps to R 2 as opposed to R in a linear functional. (c) TRUE. The trace function maps to the field and it’s linear. (d) FALSE. f (1 , , 0) = 1 but f (2 , , 0) = 4 6 = 2 · f (1 , , 0)....
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Math121sol5 - Math 121 Linear Algebra and Applications Prof...

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