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Chapter 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 \(A\xvec = \bvec\) has a solution when the columns of \(A\) span \(\real^m\text{.}\) We also found that any solution of the equation \(A\xvec = \bvec\) is unique when the columns of \(A\) are linearly independent. In this chapter, we explore the situation in which these two conditions hold simultaneously.