Vectors, matrices, and linear combinations

\(\newcommand{\avec}{{\mathbf a}} \newcommand{\bvec}{{\mathbf b}} \newcommand{\cvec}{{\mathbf c}} \newcommand{\dvec}{{\mathbf d}} \newcommand{\dtil}{\widetilde{\mathbf d}} \newcommand{\evec}{{\mathbf e}} \newcommand{\fvec}{{\mathbf f}} \newcommand{\nvec}{{\mathbf n}} \newcommand{\pvec}{{\mathbf p}} \newcommand{\qvec}{{\mathbf q}} \newcommand{\svec}{{\mathbf s}} \newcommand{\tvec}{{\mathbf t}} \newcommand{\uvec}{{\mathbf u}} \newcommand{\vvec}{{\mathbf v}} \newcommand{\wvec}{{\mathbf w}} \newcommand{\xvec}{{\mathbf x}} \newcommand{\yvec}{{\mathbf y}} \newcommand{\zvec}{{\mathbf z}} \newcommand{\rvec}{{\mathbf r}} \newcommand{\mvec}{{\mathbf m}} \newcommand{\zerovec}{{\mathbf 0}} \newcommand{\onevec}{{\mathbf 1}} \newcommand{\real}{{\mathbb R}} \newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]} \newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]} \newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]} \newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]} \newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]} \newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]} \newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]} \newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]} \newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]} \newcommand{\laspan}[1]{\text{Span}\{#1\}} \newcommand{\bcal}{{\cal B}} \newcommand{\ccal}{{\cal C}} \newcommand{\scal}{{\cal S}} \newcommand{\wcal}{{\cal W}} \newcommand{\ecal}{{\cal E}} \newcommand{\coords}[2]{\left\{#1\right\}_{#2}} \newcommand{\gray}[1]{\color{gray}{#1}} \newcommand{\lgray}[1]{\color{lightgray}{#1}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\row}{\text{Row}} \newcommand{\col}{\text{Col}} \renewcommand{\row}{\text{Row}} \newcommand{\nul}{\text{Nul}} \newcommand{\var}{\text{Var}} \newcommand{\corr}{\text{corr}} \newcommand{\len}[1]{\left|#1\right|} \newcommand{\bbar}{\overline{\bvec}} \newcommand{\bhat}{\widehat{\bvec}} \newcommand{\bperp}{\bvec^\perp} \newcommand{\xhat}{\widehat{\xvec}} \newcommand{\vhat}{\widehat{\vvec}} \newcommand{\uhat}{\widehat{\uvec}} \newcommand{\what}{\widehat{\wvec}} \newcommand{\Sighat}{\widehat{\Sigma}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

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.