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Unformatted text preview: Newberger Math 247 Spring 02 Sample Exam 2 1. a. (5 points) State the definition of one-to-one. A transformation T : R n R m is called one-to-one if for each b in R m there is at most one x in R n with T ( x ) = b . b. (5 points) State the definition of onto. A transformation T : R n R m is called one-to-one if for each b in R m there is at least one x in R n with T ( x ) = b . 2. Let T ( x 1 ,x 2 ,x 3 ) = (3 x 2 + 2 x 3 , 3 x 1- 4 x 2 ). a. (8 points) Find the standard matrix for T . T (1 , , 0) = 3 , and T (0 , 1 , 0) = 3- 4 , and T (0 , , 1) = 2 , so the standard matrix for T is A = 3 2 3- 4 0 . b. (5 points) Is T one-to-one? T is not one-to-one since there is not a pivot in every column, so, when it is consistent, the equation A x = b will have free variables and infinitely many solutions. c. (5 points) Is T onto? T is onto since there is a pivot in every row and the equation A x = b will have a solution for every b . 3. a. (8 points) If A is an invertible n n matrix, and b is a vector in R n , can A x = b have an infinite number of solutions? Why or why not?...
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- Spring '03