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Chapter 2 Vectors, Matrices, and Linear Combinations

We began our study of linear systems in Chapter 1 where we described linear systems in terms of augmented matrices, such as

\begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 2 \amp -1 \amp 3 \\ -3 \amp 3 \amp -1 \amp 2 \\ 2 \amp 3 \amp 2 \amp -1 \\ \end{array} \right] \end{equation*}

In this chapter, we will uncover geometric information in a matrix like this, which will lead to an intuitive understanding of the insights we previously gained into the solutions of linear systems.