7.1: The Syntax for Systems of Linear Equations
- Page ID
- 63900
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The following video explains the different syntax we use to describe linear systems.
The following is a summary of the syntax shown in the video:
Linear Equation
\[ b = a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + \ldots + a_{n}x_{n} \nonumber \]
System of linear equations
\[b_1 = a_{11}x_1+a_{12}x_2+a_{13}x_3 + \ldots a_{1n} \nonumber \]
\[b_2 = a_{21}x_1+a_{22}x_2+a_{23}x_3 + \ldots a_{2n} \nonumber \]
\[b_3 = a_{31}x_1+a_{32}x_2+a_{33}x_3 + \ldots a_{3n} \nonumber \]
\[\vdots \nonumber \]
\[b_m = a_{m1}x_1+a_{m2}x_2+a_{m3}x_3 + \ldots a_{mn} \nonumber \]
System of linear equations (Matrix format)
\[\begin{split}
\left[
\begin{matrix}
b_1 \\
b_2 \\
b_3 \\
\vdots \\
b_m
\end{matrix}
\right]
=
\left[
\begin{matrix}
a_{11} & a_{12} & a_{13} & & a_{1n} \\
a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\
a_{31} & a_{32} & a_{33} & & a_{3n} \\
& \vdots & & \ddots & \vdots \\
a_{m1} & a_{m2} & a_{m3} & & a_{mn}
\end{matrix}
\right]
\left[
\begin{matrix}
x_1 \\
x_2 \\
x_3 \\
\vdots \\
x_m
\end{matrix}
\right] \end{split} \nonumber \]
System of linear equations (Augmented Form)
\[\begin{split}
\left[
\begin{matrix}
a_{11} & a_{12} & a_{13} & & a_{1n} \\
a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\
a_{31} & a_{32} & a_{33} & & a_{3n} \\
& \vdots & & \ddots & \vdots \\
a_{m1} & a_{m2} & a_{m3} & & a_{mn}
\end{matrix}
\, \middle\vert \,
\begin{matrix}
b_1 \\
b_2 \\
b_3 \\
\vdots \\
b_m
\end{matrix}
\right]
\end{split} \nonumber \]