The vector equation

\begin{equation*} x_1 \vec{a}_1 + \cdots + x_n \vec{a}_n = \vec{b}, \end{equation*}

asserts that

\(\vec{b}\) is a linear combination of \(\vec{a}_1, \ldots, \vec{a_n}\)

if and only if there exists a solution to the equivalent matrix equation

\begin{equation*} A \vec{x} = \vec{b} \end{equation*}

or the equivalent linear system with the augmented matrix

\begin{equation*} \begin{pmatrix} \vphantom{\Big(} A & \vec{b} \, \end{pmatrix} = \begin{pmatrix} \vphantom{\Big(} \vec{a}_1 & \cdots & \vec{a}_n & \vec{b} \end{pmatrix}. \end{equation*}