quiz4-sol

# quiz4-sol - 1 3 x 2 3 x 1-2 x 2 = 1-2-1 3 3-2 ± x 1 x 2 ²...

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Name: MATH227 Quiz 4 (Fall 2006) For full credit, please show all steps in details!! 1. If T : R 2 R 2 is a linear transformation that rotates points (about the origin) through - π 4 radians, ﬁnd the standard matrix of T . (5 points) Since T ( e 1 ) = ± cos - π 4 sin - π 4 ² = ± 1 2 - 1 2 ² and T ( e 2 ) = ± sin - π 4 cos - π 4 ² = ± 1 2 1 2 ² , the standard matrix of T is ± 1 2 1 2 - 1 2 1 2 ² . 2. Let T : R 2 R 3 be a linear transformation such that T ± x 1 x 2 ² = x 1 - 2 x 2 - x 1 + 3 x 2 3 x 1 - 2 x 2 . Find x such that T ( x ) = - 1 4 9 . (5 points) Note that T ± x 1 x 2 ² = x 1 - 2 x 2 - x
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Unformatted text preview: 1 + 3 x 2 3 x 1-2 x 2 = 1-2-1 3 3-2 ± x 1 x 2 ² . To solve T ( x ) = -1 4 9 , reduce the augmented matrix, 1-2-1-1 3 4 3-2 9 R 2 + R 1 ,R 3-3 R 1 ∼ 1-2-1 1 3 4 12 R 3-4 R 2 ∼ 1-2-1 1 3 R 1 +2 R 2 ∼ 1 5 1 3 . Therefore, x = ± 5 3 ² . 1...
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