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Quiz5Solutions

# Quiz5Solutions - T must also be one-to-one Since T is...

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Newberger Math 247 Spring 03 Quiz 5 over Section 1.9 name: 1. Let T be the linear transformation that rotates clockwise by π/ 2 radians and then reflects across the x 1 axis. Find the standard matrix for T . Find the standard matrix by calculating T ( 1 0 ) = 0 1 , and T ( 0 1 ) = 1 0 . Then the standard matrix is 0 1 1 0 . This has a pivot in every row and every column, so it is both one-to-one and onto. 2. Let A = 3 3 3 4 2 2 and let T : R 2 R 3 be given by T ( x ) = A x . Is T one-to-one? Is T onto? How do you know? A reduces to 1 0 0 1 0 0 . It is one-to-one since it has a pivot in every column, and it is not onto since there is a row without a pivot. 3. Suppose T : R 3 R 3 is linear and onto. Explain why T must also be one-to-one.
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Unformatted text preview: T must also be one-to-one. Since T is linear, there is a 3 × 3 matrix A such that T ( x ) = A x . Since T is onto, A has a pivot in every row. But since A is 3 × 3 A also has a pivot in every column. Thus T is one-to-one. 4. Give an example of transformation that is onto but not one-to-one. Let m and n be positive integers with n > m . Let T : R n → R m be given by T ( x ) = A x where A is any m × n matrix having a pivot position in every row. Then T is onto but not one-to-one. For example, we could take T : R 3 → R 2 be given by T ( x ) = A x where A = • 1 0 0 0 1 0 ‚ ....
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