Mathematics-Online course: Preparatory Course Mathematics-Linear Algebra and Geometry-Systems of Linear Equations-Gaussian Transformation

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	Mathematics-Online course: Preparatory Course Mathematics - Linear Algebra and Geometry - Systems of Linear Equations
	Gaussian Transformation

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The following operations do not change the solution set of a LSE:

Interchange of two equations,
Multiplication of an equation by a factor $\ne 0$ ,
Subtraction of one equation from another equation.

By combination of the last two operations the addition of a multiple of one row to another row is an allowed operation.

Let's transform the linear system of equations

$\begin{displaymath} \begin{array}{rcrcrcl} &-& 2x_2 &+& 5x_3 &=& 7 \\ -8x_1 &-& 4x_2 && &=& -12 \\ 4x_1 &+& 3x_2 &+& x_3 &=& 6 \end{array}\end{displaymath}$

into triangular form using Gaussian transformations. Firstly, permute the first and the third line:

$\begin{displaymath} \begin{array}{rcrcrcl} 4x_1 &+& 3x_2 &+& x_3 &=& 6 \\ -8x_1 &-& 4x_2 && &=& -12 \\ &-& 2x_2 &+& 5x_3 &=& 7 \,. \end{array}\end{displaymath}$

Addition of the first line multiplied by 2 to the second line yields

$\begin{displaymath} \begin{array}{rcrcrcl} 4x_1 &+& 3x_2 &+& x_3 &=& 6 \\ && 2x_2 &+& 2x_3&=& 0 \\ &-& 2x_2 &+& 5x_3 &=& 7 \,. \end{array}\end{displaymath}$

Finally, addition of the second to the third line yields the triangular form

$\begin{displaymath} \begin{array}{rcrcrcl} 4x_1 &+& 3x_2 &+& x_3 &=& 6 \\ && 2x_2 &+& 2x_3&=& 0 \\ && && 7x_3 &=& 7 \,. \end{array}\end{displaymath}$

(Authors: Wipper/Abele)

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automatically generated 1/9/2017