Download Section 1: System of Linear Equations

Section 1: System of Linear Equations
Objective of this section: using a simple, efficient algorithm to solve the
system of linear equations.
Example:
In high school, you have learned the following linear equation:
(a) x  3 y  9 (b) 2 x  y  4 z  8 (c) a1 x  a2 y  b, a1 , a2 , b are constants.
Definition of a Linear Equation in n Variables:
A linear equation in n variable x1 , x2 ,, xn has the form
a1 x1  a2 x2    an xn  b ,
where the coefficients a1 , a 2 ,, an , b are real numbers (usually known). The
number of a1 is the leading coefficient and x1 is the leading variable.
In this section, we are interested in the collection of several linear equations. The
collection of these linear equations are referred to as the system of linear equations.
Definition of System of m Linear Equation in n Variables:
A system of m linear equations in n variables is a set of m equations, each of
which is linear in the same n variables:
a11x1  a12 x2    a1n xn  b1
a21x1  a22 x2    a2 n xn  b2

am1 x1  am 2 x2    amn xn  bm
where aij , bi , i  1,2,, m, j  1, n, are constants.
Example:
Consider the following system of linear equations:
x1  3x 2  x3  4
3x1  2 x 2  5 x3  2
2 x1  x 2  3x3  8
x1
 7 x3  6
 m  4, n  3