Linear Algebra (0)

A linear equation is an equation that can be written in the form 

a1x1 + a2x2 + ... + anxn = b

where a1,a2,...,an and b are real or complex numbers. A linear system is a collection of one or more linear equations involving the same variables. A solution of a system is a list s1, s2, ..., sn of numbers that make each equation a true statement when the values of s1, s2, ... , sn are substituted for x1, x2, ... , xn, respectively. The set of all possible solutions is called the solution set of the linear system. Two linear systems are equivalent if they have the same solution set.

A system of linear equations has

1.) no solution, or

2.) exactly one solution, or

3.) infinitely many solutions.

A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution.

Matrix Notation

The equations 

x    -   2y +  z  =  0

          2y -  8z =  8

-4x + 5y + 9z = -9

can be compactly recorded in the following ways. First, by using a coefficient matrix, and by using an augmented matrix.

  1   -2    1

  0    2   -8

-4     5    9

  1   -2    1    0
  0    2   -8    8
-4     5    9   -9