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Unformatted text preview: Newberger Math 247 Spring 02 Homework solutions: Section 1.5 #26-32 and Section 1.7 #23-29 Section 1.5 #26-32. 26. Suppose A x = b has a solution. Explain why the solution is unique precisely when A x = has only the trivial solution. A x = has only the trivial solution when it has a unique solution, which means there are no free variables in the system and so there is a pivot position in every column of the matrix. Similarly, A x = b has a unique solution precisely when the system has no free variables, which means the matrix has a pivot position in every column. 27. Suppose A is the 3 3 zero matrix. Describe the solution set of the equation A x = . We solve the system 0 0 0 0 0 0 0 0 0 x 1 x 2 x 3 = . There are no pivots so all three variables are free. We get x 1 x 2 x 3 = 1 x 1 + 1 x 2 + 1 x 3 , where x 1 ,x 2 and x 3 are any real numbers. Thus any vector x in R 3 is a solution to this equation. 28. If b 6 = , can the solution set of A x = b be a plane through the origin? Saying the solution set is a plane through the origin means that the origin, x 6 = , is a solution. But this means x 6 = satisfies A x = b , in other words, this means that A = b . But A = 6 = b , so is not in the solution set, and the solution set cannot be a plane through the origin. #29-32: (a) does the equation A x = have a nontrivial solution, and (b) does the equation A x = b have at least one solution for every possible b ?...
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- Spring '03