2.1 Linear system of equations

2.2 Gaussian elimination and Gauss-Jordan elimination

2.3 Exercise

A system of linear equations and its solution is one of the most important problems in Linear Algebra. A linear system with thousands of variables occurs in natural and social sciences, engineering, as well as traffic problems, weather forecasting, decision-making, etc. Even differential equations concerning derivatives such as velocity and acceleration can be solved by transforming them into a linear system.

  In Linear Algebra, a solution of a linear system is obtained by Gauss elimination method or with determinants. In Chapter 2, we consider a geometric meaning of the solution of a linear system and its solution, and investigate some applications of a linear system.


The theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics.

Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics.

In this section, we study the process of finding solutions of linear system of equations and its geometric meanings.

Equations  can be written as  and they are linear. But   are not linear.      


Describe all possible solution sets in  of a linear system with three equations and three unknowns.



One can show that there are three possibilities by a geometric method. Let us denote each equation by a plane respectively.


① It has a unique solution.

Three planes meets in a unique point.



It has infinitely many solutions.

[Ex] (1)     (2)     (3) 



③ It has no solution. (It is called 'inconsistent').


[Ex] (1)            (2) 


    (3)             (4) 



Consider matrices




 is a  matrix, and  is a  matrix, and  , and  are  matrix respectively. The main diagonal entries of  are , and  is also written as  .            



Find the augmented matrix of the following linear system of equations.




    (Find augmented matrix from a given LSE.)


 Let  be the coefficient matrix,   the unknown, and   the constant, then




Hence we have



Its augmented matrix is



Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations.

It is usually understood as a sequence of operations performed on the associated matrix of coefficients.

Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination.

Linear system of equations can be easily solved by using Gauss–Jordan elimination.


In the following procedure, the left side shows solving a linear system directly, and the right side shows solving it using an augmented matrix.


Add the multiplication of the first equation by  to the second equation.




Add the multiplication of the first equation by -3 to the third equation.



Multiply the second equation by  to get



Add the multiplication of the second equation by  to the third equation.




Multiply the third equation by .




Thus the system reduces to  

Now substituting  in the second equation, we get . Substituting and  in the first equation,

we get . Hence the solution is                          


The following are all in REF.



Consider matrices

Since matrices  do not satisfy the above properties (1), (2), (3) respectively, they are not in REF.


The following are all in RREF.

The following are equivalent.



      (Find RREF of a given matrix.)  (RREF를 구하시오.) 

  Find the RREF of .







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Solve the following by the Gauss elimination.




 Its augmented matrix is  and its REF by EROs is  .

    Therefore, since the corresponding linear system of the above augmented matrix is 




The solution is .                              


 Gauss-Jordan elimination: This is a method to transform the augmented matrix of a linear system into RREF.


Solve the following system using the Gauss-Jordan elimination.



 We will use Sage to solve this.                          

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[  1   3   0   4   2   0   0]

[  0   0   1   2   0   0   0]

[  0   0   0   0   0   1 1/3]

[  0   0   0   0   0   0   0]


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Its corresponding linear system is




 By letting ( are any real), its solution is


 From  , we can express a general solution as a vector form with any .



Using the Gauss-Jordan elimination, express the solution of the following homogeneous equation as a vector form.




 Its augmented matrix is , and its RREF is .

    Thus, leading entry 1's correspond to leading variables  and

   the rest variables  to free variables. We have the following.




Now let free variables be , then




Sec 2.2 Example 8. Solve LSE by using RREF (Gauss 소거법).


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 Answer the questions for the following linear system.



 (1) Find the coefficient matrix.

 (2)Express the linear system in the form .

 (3) Find it’s augmented matrix.



(1) The coefficient matrix of the following linear system is .

(2) Let  .

So the following linear system can be expressed by   .


(3) It’s augmented matrix is  ■

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 Find a linear system with its augmented matrix.

            (Put the unknowns as .)




Let  be the coefficient matrix,  the unknown, and b the constant.

then  =    = 

Hence we have

                               <=> =

∴ Therefore, linear system is 



This problem does not need to use Sage.  ■





 Find the number of leading variables and

free variables in the solution set of the following system.






SOL> Let   .


    =  RREF



Leading variables are and free variables are .

Double checked by Sage > http://math3.skku.ac.kr/home/pub/69 by 주영은

        okay. ■



 Which matrices are in REF or RREF? If one is not in RREF, transform it to a form in RREF.


Sol) Let the matrices .

At the matix   row’s first number(except 0) is not zero. So it is not REF and also RREF too.

At the matix B  row’s first number(except 0)is in the same column with  row. So it is not REF and RREF.


For the matrix ,


For the matrix B,


Answer : There are nor REF and RREF.

For each matrix RREF is ,.                  

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 Solve the system using Gaussian elimination.





The augmented matrix of the system can be expressed as below.





However, at , the last equation in the system is  and it is impossible. Therefore, there’s no answer for the system           ■



 Solve the system using Gauss-Jordan elimination.



Sol) The forth row is the double of the second row, so we now ignore it in this system of linear equations.





So .



We can also solve this problem by using Sage   http://math1.skku.ac.kr/home/pub/2430/    ■



 In the following circuit, write a linear system to find current .





[Kirchhoff's Law : ⓵  where  is the current 1)

                   ⓶  where  is the voltage 2)]


 (From ⓵)

   (From ⓶)




  ⇒  where 

coefficient matrix 

 of )


 Current  : .    ■

Double checked by Sage http://math1.skku.ac.kr/home/math2013/343/

        Aug =matrix([[1,-1,0,0,0],[0,1,-1,-1,0],[0,1,3,0,9],[0,1,0,6,9]])

        print Aug.rref()

        [1 0 0 0 3]

        [0 1 0 0 3]

        [0 0 1 0 2]

        [0 0 0 1 1]



 In general, we are given a linear system with  equations and  unknowns. 







If there are  free variables, what is the number of leading variables?

From this, think about the relation among the numbers of free variables, leading variables, and unknowns.




In a homogeneous linear system with n unknowns, if the RREF of the augmented matrix has k leading 1’s, the solution set has n-k free variables.

From this, we can conclude that the sum of the number of free variables and that of leading variables is a whole number of unknowns.




This problem does not need to use Sage.  ■


2015.10.11. Final OK by SGLee



Using  and , we can make equations.




Because dimension of  is 2, there are 2 linearly independent equations.








<Chapter 2 p3 NEW>

 (문제변형) Suppose that three points  pass through the parabola .

By plugging in these points, obtain three linear equations. Find coefficients  by solving 


( pass through the parabola)


         , where 




Double checked by Sage : http://math1.skku.ac.kr/home/pub/2487

(by 2009313441 김동욱)    ■



 Write a linear system with four unknowns and four equations satisfying each condition below.


 (a) A solution set with one unknown.

 (b) A solution set with two unknowns.


(a)  is an example.

Let A be the matrix . By using a Sage, RREF(A) is .

Free variable is only . Therefore, this example is a solution set with one unknown.



 is an example.

Let B be the matrix . By using a Sage, RREF(B) is .

Free variables are . Therefore, this example is a solution set with two unknowns.

Double checked by Sage> http://math3.skku.ac.kr/home/pub/70


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