Understanding Linear Algebra

In Chapter 2, we examined the two fundamental questions concerning the existence and uniqueness of solutions to linear systems independently of one another. We found that every equation of the form $A\mathbf{x}=\mathbf{b}$ has a solution when the span of the columns of $A$ is ${\mathbb{R}}^m\text{.}$ We also found that the solution $\mathbf{x}=\mathbf{0}$ of the homogeneous equation $A\mathbf{x}=\mathbf{0}$ is unique when the columns of $A$ are linearly independent. In this chapter, we explore the situation in which these two conditions hold simultaneously.