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 = ± 234 851 ² 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± 3 1 ² , ± 2 1 ²´ of IR 2 . Answer 3.1 We must fnd the matrix M = Mat A satisFying AV = WM , where V = 101 110 011 and W = ± 32 11 ² . Thus M = W - 1 AV = ± 1 - 2 - 13 ²± 2 3 4 8 5 1 ² = ± - 21 - 5 - 12 34 11 21 ² . 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 n × n de±ne their inner product by ± ² = 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 ² = Tr( X T Y ) = Tr( XY ) = Tr(( X T Y ) T ) = Tr Y T X = Tr( - YX )= - Tr( - ) which implies that ) = 0 .
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This note was uploaded on 04/18/2011 for the course EE 205A taught by Professor Laub during the Fall '10 term at UCLA.

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20105ee205A_1_2010ee205A_1_HW3_sol - Chapter 3 Linear...

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