Mathematics of Linear Algebra

There are two types of linear equation such as Homogeneous Linear Equation and Non-homogeneous Linear Equation. If b = 0 then (1) is called Homogeneous Linear Equation and if b is not equal 0 the (1) is called Non-homogeneous Linear Equation.

A finite set of equation with the variables x 1 , x 2 - - - - -x n is called a System of Linear Equation. a 1 x 1 + a 2 x 2 + - - - - - - + a n x n = b ----------------------- (1) Where a 1, a 2 - - - - - - a n and b are real constants. An equation of this form is called Linear Equation in the variables x 1 , x 2 - - - - - - x n . The variables in a linear equation are sometimes called Unknown. And there are two types of linear equation such as Homogeneous Linear Equation and Non-homogeneous Linear Equation. If b = 0 then (1) is called Homogeneous Linear Equation and if b is not equal 0 the (1) is called Non-homogeneous Linear Equation.

First Step: - Consider the following system of m linear equations (or set of m simultaneous linear equations) in n unknown x 1 , x 2 - - - - - - x n. a 11 x 1 + a 12 x 2 + - - - - - - + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + - - - - - - + a 2n x n = b 2 … …. …. ….. … … .. .. . .. . .. .. . .. .. .. .. . .. System (1) . … . .. .. . .. .. . .. .. . .. . .. . .. .. . .. . .. . .. .. a m1 x 1 + a m2 x 2 + - - - - - - + a mn x n = b m We reduce the System (1) to a simpler system as follows: Step 1: - Elimination of x 1 from the second, third……..mth equations. We may assume that the order (rule) of the equations and the order (rule) of the unknowns in each equation such that a 11 is not equal 0. The variables x 1 can then be eliminated from the second, third ………………..mth equations by subtracting. a 21 /a 11 times the first equation from the second equation a 31 /a 11 times the first equation from the third equation ..