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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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This note was uploaded on 01/12/2010 for the course MATH 247 taught by Professor F,newberger during the Spring '03 term at Stanford.
- Spring '03