Linear combinations and consistency of a linear system

Statement

Let $\mathbf v_1, \dots, \mathbf v_n$ and $\mathbf b$ be vectors in $\mathbb R^m$. Form the augmented matrix whose columns are $\mathbf v_1, \dots, \mathbf v_n$ followed by $\mathbf b$. Then $\mathbf b$ is a linear combination of $\mathbf v_1, \dots, \mathbf v_n$ if and only if the linear system with this augmented matrix is consistent.

Moreover, each solution $(c_1, \dots, c_n)$ of the system gives weights with $c_1\mathbf v_1 + \dots + c_n\mathbf v_n = \mathbf b$.