Editing Linear algebra (section)

==Linear systems==
{{Main|System of linear equations}}
A finite set of linear equations in a finite set of variables, for example, {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>n</sub>''}}, or {{math|''x'', ''y'', ..., ''z''}} is called a ''' system of linear equations''' or a '''linear system'''.<ref>{{harvtxt|Anton|1987|p=2}}</ref><ref>{{harvtxt|Beauregard|Fraleigh|1973|p=65}}</ref><ref>{{harvtxt|Burden|Faires|1993|p=324}}</ref><ref>{{harvtxt|Golub|Van Loan|1996|p=87}}</ref><ref>{{harvtxt|Harper|1976|p=57}}</ref>

Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory have been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems.

For example, let
{{NumBlk2|:|<math>\begin{alignat}{7}
2x &&\; + \;&& y             &&\; - \;&& z  &&\; = \;&& 8  \\
-3x &&\; - \;&& y             &&\; + \;&& 2z &&\; = \;&& -11 \\
-2x &&\; + \;&& y &&\; +\;&& 2z  &&\; = \;&& -3 
\end{alignat}</math>|S}}
be a linear system.

To such a system, one may associate its matrix 
:<math>M = \left[\begin{array}{rrr}
2 & 1 & -1\\
-3 & -1 & 2  \\
-2 & 1 & 2
\end{array}\right].
</math>
and its right member vector
:<math>\mathbf{v} = \begin{bmatrix} 8\\-11\\-3 \end{bmatrix}. </math>

Let {{mvar|T}} be the linear transformation associated with the matrix {{mvar|M}}. A solution of the system ({{EquationNote|S}}) is a vector 
:<math>\mathbf{X}=\begin{bmatrix} x\\y\\z \end{bmatrix}</math> 
such that 
:<math>T(\mathbf{X}) = \mathbf{v},</math>
that is an element of the [[preimage]] of {{mvar|v}} by {{mvar|T}}.

Let ({{EquationNote|S′}}) be the associated [[Homogeneous system of linear equations|homogeneous system]], where the right-hand sides of the equations are put to zero:

{{NumBlk2|:|<math>\begin{alignat}{7}
2x &&\; + \;&& y             &&\; - \;&& z  &&\; = \;&& 0  \\
-3x &&\; - \;&& y             &&\; + \;&& 2z &&\; = \;&& 0 \\
-2x &&\; + \;&& y &&\; +\;&& 2z  &&\; = \;&& 0 
\end{alignat}</math>|S′}}

The solutions of ({{EquationNote|S′}}) are exactly the elements of the [[kernel (linear algebra)|kernel]] of {{mvar|T}} or, equivalently, {{mvar|M}}.

The [[Gaussian elimination|Gaussian-elimination]] consists of performing [[elementary row operation]]s on the [[augmented matrix]]
:<math>\left[\!\begin{array}{c|c}M&\mathbf{v}\end{array}\!\right] = \left[\begin{array}{rrr|r}
2 & 1 & -1&8\\
-3 & -1 & 2&-11  \\
-2 & 1 & 2&-3
\end{array}\right]
</math>
for putting it in [[reduced row echelon form]]. These row operations do not change the set of solutions of the system of equations. In the example, the reduced echelon form is 
:<math>\left[\!\begin{array}{c|c}M&\mathbf{v}\end{array}\!\right] = \left[\begin{array}{rrr|r}
1 & 0 & 0&2\\
0 & 1 & 0&3  \\
0 & 0 & 1&-1
\end{array}\right],
</math>
showing that the system ({{EquationNote|S}}) has the unique solution
:<math>\begin{align}x&=2\\y&=3\\z&=-1.\end{align}</math>

It follows from this matrix interpretation of linear systems that the same methods can be applied for solving linear systems and for many operations on matrices and linear transformations, which include the computation of the [[rank of a matrix|ranks]], [[kernel (linear algebra)|kernels]], [[matrix inverse]]s.