Editing System of linear equations (section)

===Vector equation===
One extremely helpful view is that each unknown is a weight for a [[column vector]] in a [[linear combination]].
:<math>
x_1\begin{bmatrix}a_{11}\\a_{21}\\ \vdots \\a_{m1}\end{bmatrix} +
x_2\begin{bmatrix}a_{12}\\a_{22}\\ \vdots \\a_{m2}\end{bmatrix} +
\dots +
x_n\begin{bmatrix}a_{1n}\\a_{2n}\\ \vdots \\a_{mn}\end{bmatrix} = \begin{bmatrix}b_1\\b_2\\ \vdots \\b_m\end{bmatrix}
</math>
This allows all the language and theory of ''[[vector space]]s'' (or more generally, ''[[Module (mathematics)|modules]]'') to be brought to bear. For example, the collection of all possible linear combinations of the vectors on the [[Sides of an equation|left-hand side]] (LHS) is called their ''[[Span (linear algebra)|span]]'', and the equations have a solution just when the right-hand vector is within that span. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. In any event, the span has a ''[[basis (linear algebra)|basis]]'' of [[linearly independent]] vectors that do guarantee exactly one expression; and the number of vectors in that basis (its ''[[dimension (linear algebra)|dimension]]'') cannot be larger than ''m'' or ''n'', but it can be smaller. This is important because if we have ''m'' independent vectors a solution is guaranteed regardless of the right-hand side (RHS), and otherwise not guaranteed.