Equivalence with linear combination

The vector equation

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

asserts that

if and only if a 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

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