Gauss-Jordan Elimination

Gauss-Jordan elimination is a method of elimination, named after Carl Gauss and Willhelm Jordan. It continues the reduction process until a reduced echelon-form is obtained.

Example:
Use Gauss Jordan Elimination to solve the system.
           
                                        3x - 5y - 6z = 16
                                        4x - 8y        = 12
                                        2x + 4y + 8z =24

                                     
                                        [ 3   -5   -6   16 ]
                                        [ 4   -8    0    12 ]                               1/3(R1) ---> R1
                                        [ 2    4    8    24 ]

                                        [ 1  -5/3  -2   16/3 ]
                                        [ 4    -8     0     12 ]                     -4(R1) + R2 ---> R2
                                        [ 2     4     8      24 ]

                                        [ 1  -5/3   -2  16/3 ]
                                        [ 0  -4/3   8  -28/3 ]                    -2(R1) + R3 ---> R3 
                                        [ 2     4     8      24 ]
       
                                        [ 1  -5/3     -2      16/3]
                                        [ 0  -4/3      8     -28/3]                    -4/3(R2) ---> R2     
                                        [ 0  22/3     12     40/3]

                                        [ 1   -5/3    -2      16/3]
                                        [ 0      1      -6       7   ]        -22/23(R2) + R3 ---> R3
                                        [ 0   22/3    12    40/3]    

                                        [ 1   -5/3    -2       16/3]
                                        [ 0     1       -6        7   ]                   1/56(R3) ---> R3
                                        [ 0     0      56      -38  ]

                                        [ 1   -5/3    -2       16/3]
                                        [ 0     1       -6        7   ]               6(R3) + R2 ---> R2
                                        [ 0    0         1   -19/28]

                                        [ 1   -5/3     -2       16/3]
                                        [ 0      1        0     41/14]              2(R3) + R1 ---> R1   
                                        [ 0      0        1  - 19/28]

                                        [ 1   -5/3       0  167/42]
                                        [ 0      1         0   41/14]            5/3(R2) + R1 ---> R1
                                        [ 0      0         1  -19/28]  

                                        [ 1      0         0     62/7]           a=   62/7
                                        [ 0      1         0   41/14]           b=   41/14
                                        [ 0      0         1  -19/28]           c= -19/28

Elementary Row Operations

There are three operations that correspond to Elementary Row Operations. 

1. Interchange two equations.
   
Example:
a. Interchange the first and second rows.  
                Original Matrix             New Row-Equivalent Matrix
                  [ 0   1   7   2 ]                         [ 1   4   2   3 ]
                  [ 1   4   2   3 ]                         [ 0   1   7   2 ]           R1 <---> R2
                  [ 3   7  -1   3 ]                         [ 2  -3   4   1 ]

2. Multiply an equation by a nonzero constant.
                
Example:
b. Multiply the first row by -2 in order to produce a new first row.              
                 Original Matrix            New Row-Equivalent Matrix
                   [ 1   6  -3   2]                          [-2  -12  6  -4 ]             
                   [ 2   4  -4   0]                          [ 2     4  -4   0 ]         -2(R1) ---> R1
                   [ 6   2  -8   1]                          [ 6     2  -8   1 ]

3. Add a multiple of an equation to another equation.

Example:
c. Add -4 times times the first row to the third row to produce a new third row.
                 Original Matrix             New Row-Equivalent Matrix
                   [ 2   4   2   1 ]                         [ 2   4   2   1 ]
                   [ 3   1   9   4 ]                         [ 3   1   9   4 ]            R3 + R1(-4) ---> R3
                   [ 8   8    3   6 ]                        [ 0  -8  -5   2 ]

*Two matrices are row-equivalent if one can be obtained from the other by a finite sequence of elementary row operations.

Augmented and Coefficient Matrices

An Augmented Matrix is a matrix derived from the coefficients and constant terms of a system of linear equations, while a Coefficient Matrix is a matrix containing only the coefficients of the system.

Here is an example:

     System                                Augmented Matrix           Coefficient Matrix

5x - 9y + 2z = 5                           [ 5    -9    2   5 ]                  [ 5   -9   2 ]

3x + 7y - 2z = 9                           [ 3     7   -2   9 ]                  [ 3    7  -2 ]

x            + 9z = 20                       [1      0    9  20]                  [ 1    0    9 ]

In the example, the Augmented Matrix have both the coefficients and their constant terms as entries, while the coefficient matrix only have the coefficients.

Solving using Back Substitution

Sample Equation:

10x + 9y – z = 0
      15y – 5z = 20
                   z = 2

It is fairly easy to see how to proceed in this case. I'll just back-substitute the z-value from the third equation into the second equation, solve the result for y, and then plug z and y into the first equation and solve the result for x.

15y – 5(2) = 20                  
15y – 10 = 20
15y = 30

Therefore, y = 2

10x + 9(2) – (3) = 0
10x + 18 – 3 = 0
5x + 15 = 0
5x = –15

Therefore, x = -3

Then the solution is:
x =-3
y = 2
z = 2

The reason this system was easy to solve is that the system was "triangular"; this refers to the equations having the form of a triangle, because of the lower equations containing only the later variables.

Linear Algebra

What is Linear Algebra?

Linear algebra is the study of linear sets of equations and their transformation properties. It allows the analysis of rotations in space,least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering. 

It is also used in vector spaces and linear mappings between such spaces. It is also the study of lines, planes, and subspaces and their intersections using algebra. Linear algebra assigns vectors as the coordinates of points in a space, so that operations on the vectors define operations on the points in the space

Why should everyone study it?

Linear algebra is said to be vital in multiple areas of science. Practically every area of modern science contains models where equations are approximated by linear equations and solving for the system helps the theory develop. It is also vital in Engineering, because it allows you to manipulate and understand whole systems of equations with huge numbers of dimensions/variables on paper without any fuss, and solve them computationally.

History of Linear Algebra
The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. They were used by Leibniz in 1693, while Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination.

Introduction

"Life is good for only two things, discovering mathematics and teaching mathematics."

- Simeon Poisson

Hello! I am Jonell Tanagras, 15 years old, a 4^th year student  in Regional Science High School III. This blog is about me and my studies on Linear Algebra. I will change your impression about this by posting different lessons that I had studied every Friday. Let me prove to you that Linear Algebra isn't that hard.