Solutions Manual to accompany An Introduction to Numerical Methods and Analysis, 3rd Edition

CHAPTER 7NUMERICAL METHODS FOR THE SOLUTION OF SYSTEMS OF EQUATIONS

7.1 LINEAR ALGEBRA REVIEW

Exercises:

1. Assume Theorem 7.1 and use it to prove Cor. 7.1.

   Solution: If is singular, then Part 2 of the theorem says that the columns of are a dependent set. Therefore, there exists a set of constants , , , , not all zero, such that

   

   where the are the columns of written as vectors. But this is equivalent to saying , where has components . Therefore there exists at least one non‐zero vector ...