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Unformatted text preview: Solutions to problems from section 1.9 For questions 2, 4, 10 assume that T is a linear map and find the standard matrix A for T . [Recall that by theorem 10 we know the columns of A will be the images of the e i ’s under T .] 2. T : R 3 → R 2 , T ( e 1 ) = bracketleftbigg 1 3 bracketrightbigg , T ( e 2 ) = bracketleftbigg 4 − 7 bracketrightbigg , and T ( e 3 ) = bracketleftbigg − 5 4 bracketrightbigg where e 1 , e 2 , e 3 are the columns of the 3 × 3 identity matrix. Solution: A = bracketleftbigg 1 4 − 5 3 − 7 4 bracketrightbigg 4. T : R 2 → R 2 rotates points (about the origin) through − π/ 4 radians (clockwise). Solution: We must find the images of e 1 = bracketleftbigg 1 bracketrightbigg and e 2 = bracketleftbigg 1 bracketrightbigg under T . Well, let’s draw a picture: Thus A = √ 2 2 √ 2 2 − √ 2 2 √ 2 2 10. T : R 2 → R 2 first reflects points through the vertical x 2axis and then rotates points π/ 2 radians. Solution: Again, let’s look at the picture: Thus A = bracketleftbigg − 1 − 1 bracketrightbigg 16. Fill in the missing entries of the matrix assuming that the equation holds for all values of the variables....
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 Fall '08
 AXENOVICH
 Linear Algebra, Linear map, linear transformation, standard matrix, Identity function

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