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Contents Index Dark Mode Prev Up Next \(\newcommand{\avec}{{\vec{a}}}
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\)
Chapter 3 Invertibility, bases, and coordinate systems
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
\(A\vec{x} = \vec{b}\) has a solution when the span of the columns of
\(A\) is
\(\real^m\text{.}\) We also found that the solution
\(\vec{x}=\vec{0}\) of the homogeneous equation
\(A\vec{x} = \vec{0}\) is unique when the columns of
\(A\) are linearly independent. In this chapter, we explore the situation in which these two conditions hold simultaneously.
