matrix equation

$A\mathbf x = \mathbf b$ — a whole linear system collapsed into one line.

Given an $m\times n$ coefficient matrix $A$ and target $\mathbf b\in\mathbb R^m$ , the matrix equation $A\mathbf x=\mathbf b$ asks for $\mathbf x\in\mathbb R^n$ satisfying it — exactly a linear system in disguise. By the column picture, $A\mathbf x=x_1\mathrm{col}_1(A)+\cdots+x_n\mathrm{col}_n(A)$ , so the equation asks whether $\mathbf b$ can be written as a linear combination of $A$ 's columns; it is consistent exactly when $\mathbf b$ lies in $A$ 's column space.