A [[Linear]] combination of some [[Ordered Set]] of objects $S$ (*ex.* a [[Vector]]) upon some [[Linear]] [[Binary Operation]] is the sum:
$\huge \begin{align}
x_{1}s_{1}+x_{2}s_{2}+\cdots + x_{{k}}s_{k}
\end{align}
$
For some $x_{1}, \dots, x_{k} \in \R$.
>[!example] [[Vector]] [[Linear Combination]]
>$\huge \begin{align}
>\let \vec{v_{1}},\vec{v_{2}}, \dots,\vec{v_{k}} &\in \R^n \\
>\let x_{1}, x_{2}, x_{k} &\in \R \\
> \vec{v_{1}}x_{1} +\vec{v_{2}}x_{2} + \dots +\vec{v_{k}}x_{k}
> &=c
>\end{align}$
A [[System of Linear Equations]] can always be expressed as a [[Linear Combination]] of [[Vector|Vectors]]
>[!example]-
>$\huge \begin{align}
> v_\ang{1,1}x_{1}+v_\ang{2,1}x_{2}+v_\ang{3, 1}x_{3} &= c_{1}\\
> v_\ang{1,2}x_{1}+v_\ang{2,2}x_{2}+v_\ang{3, 2}x_{3} &= c_{1}\\
>v_\ang{1,3}x_{1}+v_\ang{2,3}x_{2}+v_\ang{3, 3}x_{3} &= c_{1} \\
>\sim \vec{v_{1}}x_{1} + \vec{v_{2}}x_{2}+\vec{v_{3}}x_{3} &=\vec c & \end{align}$
$\huge \let \vec{v_{1}},\dots,\vec{v_{m}}, \vec b \in \R^{n} $
$\huge \let x_{1}, \dots, x_{{m} }\in \R $
To solve the [[Linear Combination]] for $x_{1}, \dots, x_{{m}}$
$\huge x_{1}\vec{v_{1}} + \dots +x_{m}\vec {v_{m}}=\vec b
\sim
\augmented{c|c}{
\vec{v_{1} } \cdots \vec{v_{m}} & \vec b
}$