Editing System of linear equations (section)

==General form==
A general system of ''m'' linear equations with ''n'' [[Variable (mathematics)|unknowns]] and [[coefficient]]s can be written as
:<math>\begin{cases}
a_{11} x_1 + a_{12} x_2 +\dots + a_{1n} x_n = b_1 \\
a_{21} x_1 + a_{22} x_2  + \dots + a_{2n} x_n = b_2 \\ 
\vdots\\
a_{m1} x_1 + a_{m2} x_2 + \dots + a_{mn} x_n = b_m,
\end{cases}</math>
where <math>x_1, x_2,\dots,x_n</math> are the unknowns, <math>a_{11},a_{12},\dots,a_{mn}</math> are the coefficients of the system, and <math>b_1,b_2,\dots,b_m</math> are the constant terms.{{sfnp|Beauregard|Fraleigh|1973|p=65}}

Often the coefficients and unknowns are [[Real number|real]] or [[complex number]]s, but [[integer]]s and [[rational number]]s are also seen, as are polynomials and elements of an abstract [[algebraic structure]].

===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.

===Matrix equation===
The vector equation is equivalent to a [[matrix (mathematics)|matrix]] equation of the form
<math display="block"> A\mathbf{x} = \mathbf{b} </math>
where ''A'' is an ''m''×''n'' matrix, '''x''' is a [[column vector]] with ''n'' entries, and '''b''' is a column vector with ''m'' entries.{{sfnp|Beauregard| Fraleigh|1973|pp=65&ndash;66}}

<math display="block">
A=
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix},\quad
\mathbf{x}=
\begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n
\end{bmatrix},\quad
\mathbf{b}=
\begin{bmatrix}
b_1 \\
b_2 \\
\vdots \\
b_m
\end{bmatrix}.
</math>
The number of vectors in a basis for the span is now expressed as the ''[[rank (linear algebra)|rank]]'' of the matrix.