Matrix equation $Ax=b$, its solution set, consistency, homogeneous systems and the augmented matrix $[A\mid b]$

Let $A\in M_{m\times n}(F)$ and let $b\in F^m$, viewed as an $m\times1$ column. The matrix equation $Ax=b$ asks for columns $x\in F^n$ satisfying the displayed equality. Equivalently it is the linear system $\sum_{j<n}a_{ij}x_j=b_i\qquad(i<m).$

Its solution set is $S(A,b):=\{x\in F^n:Ax=b\}$. The system is consistent when $S(A,b)\ne\varnothing$ and inconsistent otherwise. It is homogeneous when $b=0$.

The augmented matrix is $[A\mid b]\in M_{m\times(n+1)}(F)$, obtained by adjoining $b$ as the last column. Elementary row operations act on every entry of a row, including its augmented entry.