Introductions

Linear Combinations

A linear combination of $x_{1}, x_{2}, \dots, x_{n}$ is an expression

a_{1} x_{1} + a_{2} x_{2} + \dots + a_{n} x_{n}

where $x_{i}$ 's are indeterminates or variables; and $a_{i}$ 's are coefficients and belong to a field $𝕂$

\begin{matrix} a_{1 1} x_{1} + a_{1 2} x_{2} + \dots + a_{1 n} x_{n} & = & b_{1} \\ a_{2 1} x_{1} + a_{2 2} x_{2} + \dots + a_{2 n} x_{n} & = & b_{2} \\ ⋮ \\ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} & = & b_{m} \end{matrix}