3: Invertibility, bases, and coordinate systems
In Chapter 2 , we examined our two fundamental questions, Question 1.4.2 , concerning the existence and uniqueness of solutions to linear systems independently of one another. We found that every equation of the form Ax=b has a solution when the columns of A span .Rm. We also found that any solution of the equation Ax=b is unique when the columns of A are linearly independent. In this chapter, we explore the situation in which these two conditions hold simultaneously.