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20105ee205A_1_2010ee205A_1_HW3_sol

# 20105ee205A_1_2010ee205A_1_HW3_sol - Chapter 3 Linear...

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Chapter 3 Linear Transformations 1. Let A = 2 3 4 8 5 1 and consider A as a linear transformation map- ping IR 3 to IR 2 . Find the matrix representation of A with respect to the bases 1 1 0 , 0 1 1 , 1 0 1 of IR 3 and 3 1 , 2 1 of IR 2 . Answer 3.1 We must find the matrix M = Mat A satisfying AV = WM , where V = 1 0 1 1 1 0 0 1 1 and W = 3 2 1 1 . Thus M = W - 1 AV = 1 - 2 - 1 3 2 3 4 8 5 1 1 0 1 1 1 0 0 1 1 = - 21 - 5 - 12 34 11 21 . 2. Consider the vector space IR n × n over IR and let S denote the sub- space of symmetric matrices and R the subspace of skew-symmetric matrices. For matrices X, Y IR n × n define their inner product by X, Y = Tr( X T Y ). Show that with respect to this inner product, R = S . 9

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10 CHAPTER 3. LINEAR TRANSFORMATIONS Answer 3.2 Let X be an arbitrary symmetric matrix and Y be an ar- bitrary skew-symmetric matrix. The result essentially follows by noting that X, Y = Tr( X T Y ) = Tr( XY ) = Tr(( X T Y ) T ) = Tr Y T X = Tr( - Y X ) = - Tr( Y X ) = - Tr( XY ) which implies that Tr( XY ) = 0 .
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20105ee205A_1_2010ee205A_1_HW3_sol - Chapter 3 Linear...

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