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Unformatted text preview: Solutions to problems from section 1.8 4. With T defined by T ( x ) = A x , find a vector x whose image under T id b and determine whether x is unique. A = 1 − 3 2 1 − 4 3 − 5 − 9 , b = 6 − 7 − 9 Solution: We are trying to determine existence and uniqueness of a solution to the equation A x = b . Wo look at the corresponding augmented matrix: 1 − 3 2 6 1 − 4 − 7 3 − 5 − 9 − 9 NR 3 = R 3 3 R 1 −→ 1 − 3 2 6 1 − 4 − 7 4 − 15 − 27 NR 3 = R 3 4 R 2 −→ 1 − 3 2 6 1 − 4 − 7 1 1 Notice that the corresponding linear system is consistent but has no free variables. Thus there exists a vector x whose image under T id b and that x is unique. 10. Find all x in R 4 that are mapped to the zero vector by the matrix transformation x mapsto→ A x where A = 1 3 9 2 1 0 3 − 4 0 1 2 3 − 2 3 0 5 Solution: We are looking for the solution set of the equation A x = . Let’s reduce the corresponding augmented matrix: 1 3 9 2 0 1 0 3 − 4 0 0 1 2 3 0 − 2 3 0 5 0 NR 2 = R 2 R 1 NR 4 = R 4 +2 R 1 −→ 1 3 9 2 0 − 3 − 6 − 6 0 1 2 3 0 9 18 9 0 R 2 ↔ R 3 −→ 1 3 9 2 0 1 2 3 0 − 3 − 6 − 6 0 9 18 9 0 NR 3 = R 3 +3 R 2 NR 4 = R 4 9 R 2 −→ 1 3 9 2 0 0 1 2 3 0 0 0 0 3 0 0 0 0 − 18 0 NR 3 = 1 3 R 3 −→ 1 3 9 2 0 0 1 2 3 0 0 0 0 1 0 0 0 0 − 18 0 NR 1 = R 1...
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This note was uploaded on 01/28/2010 for the course MATH 307 taught by Professor Axenovich during the Fall '08 term at Iowa State.
 Fall '08
 AXENOVICH

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