Definition:Solution

Consider the system of $$m$$ simultaneous linear equations in $$n$$ variables $$x_1, x_2, \ldots, x_n$$:

$$\forall i \in \left[{1 \,. \, . \, m}\right] : \sum \limits_{j=1}^n {\alpha_{i j} x_j} = \beta_i$$

where $$\alpha_{i j}, \beta_i \in K$$ where $$K$$ is a field.

An $n$-tuple $$\left({x_1, x_2, \ldots, x_n}\right)$$ which satisfies each of the equations in a system of $$m$$ linear equations in $$n$$ variables is called a solution of the system.