Mathguru – Help – Solving linear equations using matrix method.

4.6 Applications of Determinants and Matrices > No. 11

Solving linear equations using matrix method.

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Explanation:

System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. For example,

$Description: \begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}$

is a system of three equations in the three variables $x$ , $y$ , $z$ . A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by

$Description: \begin{alignat}{2} x & = & 1 \\ y & = & -2 \\ z & = & -2 \end{alignat}$

since it makes all three equations valid.

General form

A general system of m linear equations with n unknowns can be written as

$Description: \begin{alignat}{7} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = \;&&& b_1 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = \;&&& b_2 \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && &&& \;\vdots \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = \;&&& b_m. \\ \end{alignat}$

Here $Description: x_1,\ x_2,...,x_n$ are the unknowns, $Description: a_{11},\ a_{12},...,\ a_{mn}$ are the coefficients of the system, and $Description: b_1,\ b_2,...,b_m$ are the constant terms.

Matrix equation

The vector equation is equivalent to a matrix equation of the form

$Description: A\bold{x}=\bold{b}$

where A is an m×n matrix, X is a column vector with n entries, and b is a column vector with m entries.

$Description: A= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix},\quad \bold{x}= \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix},\quad \bold{b}= \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}$

The number of vectors in a basis for the span is now expressed as the rank of the matrix.

(Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/System_of_linear_equations#Other_methods

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.