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H54 - Math 330 Homework 5.4(Pages 333,334(4 Let B = cfw_b1...

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Math 330 Homework 5.4 (Pages 333,334) (4) Let B = { vector b 1 , vector b 2 , vector b 3 } be a basis for a vector space V and T : V R 2 a linear transformation with the property that T ( x 1 vector b 1 + x 2 vector b 2 + x 3 vector b 3 ) = bracketleftBigg 2 x 1 - 4 x 2 + 5 x 3 - x 2 + 3 x 3 bracketrightBigg Find the matrix for T relative to B and the standard basis for R 2 SOLUTION: Our matrix with respect to the two bases is essentially a table where the rows describe the output basis for R 2 and the columns the input basis for V . In what appears below we let vector epsilon1 1 = bracketleftBigg 1 0 bracketrightBigg and vector epsilon1 2 = bracketleftBigg 0 1 bracketrightBigg denote the two elements for the standard basis of R 2 . T vector b 1 vector b 2 vector b 3 vector epsilon1 1 2 -4 5 vector epsilon1 2 0 -1 3 The table describes the following facts in a very compact way: With x 1 = 1 and x 2 = x 3 = 0 we have that T ( vector b 1 ) = 2 vector epsilon1 1 + 0 vector epsilon1 2 = bracketleftBigg 2 0 bracketrightBigg .
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