7.1: La Sintaxis para Sistemas de Ecuaciones Lineales
- Page ID
- 115566
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)El siguiente video explica las diferentes sintaxis que utilizamos para describir los sistemas lineales.
El siguiente es un resumen de la sintaxis que se muestra en el video:
Ecuación Lineal
\[ b = a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + \ldots + a_{n}x_{n} \nonumber \]
Sistema de ecuaciones lineales
\[b_1 = a_{11}x_1+a_{12}x_2+a_{13}x_3 + \ldots a_{1n} \nonumber \]
\[b_2 = a_{21}x_1+a_{22}x_2+a_{23}x_3 + \ldots a_{2n} \nonumber \]
\[b_3 = a_{31}x_1+a_{32}x_2+a_{33}x_3 + \ldots a_{3n} \nonumber \]
\[\vdots \nonumber \]
\[b_m = a_{m1}x_1+a_{m2}x_2+a_{m3}x_3 + \ldots a_{mn} \nonumber \]
Sistema de ecuaciones lineales (Formato Matriz)
\ [\ begin {split}
\ left [
\ begin {matrix}
b_1\\
b_2\\
b_3\\
\ vdots\\
b_m
\ end {matrix}
\ right]
=
\ left [
\ comenzar {matriz}
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}
\ final {matriz}
\ derecha]
\ izquierda [
\ comenzar {matriz}
x_1\
x_2\\
x_3\
\ vdots\\
x_m
\ end {matriz}
\ derecha]\ end {split}\ nonumber\]
Sistema de ecuaciones lineales (Forma Aumentada)
\ [\ 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_ {33} y a_ {33} & a_ {3n}\\
&\ vdots & &\ ddots &\ vdots\\
a_ {m1} & a_ {m2} & a_ {m3} & a_ {mn}
\ end {matrix}
\,\ middle\ vert\,
\ begin {matriz}
b_1\\
b_2\\
b_3\\
\ vdots\\
b_m
\ end {matrix}
\ right]
\ end {split}\ nonumber\]