Linear Algebra with Applications (3rd Edition)

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Math 417 homework 1 solutions 1 Solutions Most of the questions are straightforward computation; contact the instructor if you still have questions. Section 1.3 problem 24 Let A R 4 × 4 , b R 4 . Assume that Ax = b has a unique solution x . If we change b , do we still have a unique solution? Answer: Yes, no matter what the new b is. Form the augmented matrix A . . . b and use elementary row operations to bring it to reduced row echelon form. The elementary row operation have an important property: they do not change the set of solutions x . Consider the case where the reduced row echelon form is 1 0 0 * . . . * 0 1 0 * . . . * 0 0 1 * . . . * 0 0 0 0 . . . 1 (a * can be any number). Then there is no solution, because the last row corresponds to the equation 0 = 1. (Reread notes or book to recall how to read solution sets off the r.r.e.f.) If we have 1 0 0 * . . . * 0 1 0 * . . . * 0 0 1 * . . . * 0 0 0 0 . . . 0 1

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then x 4 is a free variable, so we have infinitely many solutions.
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