Systems of linear equations

A set of equations having the form
$\left. \begin{array}{ccc} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n & = & r_1 \\ a_{21}x_2 + a_{22}x_2 + \cdots + a_{2n}x_n & = & r_2 \\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots & \cdots & \cdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n & = & r_n \end{array} \right\}\cdots(16)$
is called a system of m linear equations in the n unknowns $x_1,\ x_2,\ \cdots,\ x_n$ . If $r_1,\ r_2,\ \cdots,\ r_n$ are all zero the system is called homogeneous. If they are not all zero it is called non-homogeneous. Any set of numbers $x_1,\ x_2,\ \cdots,\ x_n$ which satisfies (16) is called a solution of the system.

In the matrix form (16) can be written
$\displaystyle \left( \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \\ \cdots \\ x_n \end{array} \right) = \left( \begin{array}{c} r_1 \\ r_2 \\ \cdots \\ r_n \end{array} \right) \cdots (17)$
or more briefly $AX = R \cdots (18)$

where $A$ , $X$ , $R$ represent the corresponding matrices in (17).