2.3 Linear equations

A system of linear equations in the variables \(x_1,\ldots,x_m\) is a list of simultaneous equations \[\begin{align} a_{11}x_1 + a_{12} x_2 + \cdots + a_{1m}x_m &= b_1 \\ a_{21}x_1 + a_{22} x_2 + \cdots + a_{2m}x_m &= b_2 \\ \vdots & \vdots \\ a_{n1}x_1 + a_{n2} x_2 + \cdots + a_{nm}x_m &= b_n \end{align}\] \tag{2.1} where the \(a_{ij}\) and \(b_i\) are numbers.

If we let \(A\) be the matrix \((a_{ij})\), \(\mathbf{b}\) the column vector \(\begin{pmatrix} b_1\\ \vdots \\ b_n \end{pmatrix}\), and \(\mathbf{x}\) the column vector \(\begin{pmatrix} x_1 \\ \vdots \\ x_m \end{pmatrix}\) then we can express (2.1) by saying

\[ \tag{2.2} A \mathbf{x} = \mathbf{b}. \]

A solution of this system is a list of values for the \(x_i\)s such that when we substitute them into (2.2), the equation holds. A system of linear equations is consistent if it has a solution, otherwise it is inconsistent.