LA_1. Linear systems

nontrivial solution (nghiệm không tầm thường) then it has infinite family of solutions (vô số nghiệm).. Hệ thuần nhất mà số ẩn nhiều hơn số pt[r]

OUR GOAL

• Elementary Operations • Gaussian Elimination

• a1x1+ a2x2+…+anxn=b

is called a linear equation (phương trình tuyến tính)

• If a1s1+a2s2+…+ansn= b



(s1,s2,…,sn) is called solution of the equation

• A system may have:

no solution

unique solution

an infinite family of solutions

coefficients variables = unknowns

1.1 Solutions and Elementary Operations

0

1

Inconsistent (không tương thích)

Consistent (tương thích)

No solutions

Example 1

•

Inconsistent Consistent

(infinitely many solutions)

(0,2,1), (2,0,1) (t,2-t,1) no solution

(t,2-t,1) is called a general solution and given in parametric form, t is parameter ( t is arbitrary)

2 1

2 3

x y

x y

 

 

 



1 3

x y z x y z

  

 

  

Algebraic Method

constant matrix augmented matrix

coefficient matrix

1

1

1

3

x x x x

x x x

x x x x

   

  

   

3 1

2

3

                       

3 2 1 1 2 0 1 2 3 1 2 5

Example

• Consider the system

augmented matrix

1

2 2

x y

x y

 

 

 



1 1

1 2 2



Solution (0,-1)

Example

1

2 3

x y x y

x y x y

              

1 1 1 1 2 3

     1 x y x y        

1 0 1   0 x y x y        

1 1

0 1

 

A row-echelon matrix has properties

• All the zero rows are at the bottom

• The first nonzero entry

from the left in each

nonzero row is a 1, called the leading 1 for that row

• Each leading is to the right of all leading 1’s in the rows above it

0 * * * * *

0 0 * * *

0 0 0 * *

0 0 0

0 0 0 1

1

1

1

0

0 0

 

 

 

 

 

 

 

Row-echelon matrix

( for any choice in *-position )

The row-echelon matrix has the “staircase” form

Which is a row-echelon matrix?

1 * * * 0

 

 

 

 

 

1 * * 0

 

 

 

1 * * * * * * *

 

 

 

 

 

0 * * 0 0 0

 

 

 

 

 

1 * * 0 0

 

 

 

 

A reduced row-echelon matrix

(ma trận bậc thang theo dòng thu gọn) has the properties

• It is a row-echelon matrix • Each leading is the only

nonzero entry in its column

1

0 1

0 0 1

0

*

0

*

0

 

 

 

 

Which is a reduced row- echelon matrix?

1 * 0 0

 

 

 

 

 

1 * 0

 

 

 

1 * 0 * 0 0

 

 

 

 

 

0 * 0 0 0 0

 

 

 

 

 

1 * 0 0 0

 

 

 

 

How to carry a matrix to

Elementary Operations (phép biến đổi sơ cấp)

• Interchange two equations (type I)

• Multiply one equation by a nonzero number (type II)

Gaussian Algorithm

• Step If all row are zeros, stop

• Step Otherwise, find the first column from the left containing a nonzero entry (call it a) and move the row containing a to the top position

• Step Multiply that row by 1/a to creat the leading 1

• Step By subtracting multiples of that row from the rows below it, make each entry below the leading

zero

• Step Repeat step 1-4 on the matrix consisting of the

remaining rows

Gaussian Algorithm

step 2 step 3 step 4 step 5

leading one 4 1 0 3

1

3

r r                     

0 0 0 0 1:

0 0 0 0

step stop              1 1 0 3

7 1 r                   

0

2

0

4              

2

0

0

2

4

rr

Example

Carry the matrix

• to row-echelon matrix • to reduced

row-echelon matrix

2 6 2 2

2 3 11 4

3 11 3 0



 

   

 

 

row-echelon matrix

3

7 1

0 0

r              1

2 2 1 11 11 11 3 11

r                          

3 1

0 6

r r r r

              

3 1

0 2

r   

         

2 1

0 0

reduced row-echelon matrix

3

2

2

3

1 3 1 1 1 3 1 0

0 1 3 2 0 1 3 0

0 0 0 1 0 0 0 1

1 0 10 0

0 1 3 0

0 0 0 1

r r r r

r r

 

 

 

 

   

    

   

   

   



 

 



 

 

The rank of a matrix

• The rank of the matrix A, rankA = the

number of leading ones in the reduced

row-echelon form of A

Rank 2

1 * 0 * 0 0

 

 

 

 

Gauss-Jordan Elimination

(for solving a system of linear equations)

• Step 1.Using elementary row

operations, augmented matrix 

reduced row-echelon matrix

• Step If a row [0 0…0 1] occurs,

the system is inconsistent

• Step Otherwise, assign the

nonleading variables as

parameters, solve for the leading variables in terms of parameters

reduced row echelon matrix

z is nonleading variable

z=t (parameter)

0 10

0 0 0

0

1 10

0 0

1

0

1

x y z x y z x y z

                         

0 10

0

0 0 0

0

0 0

1

1

0 x y z

x y z x y z

inconsistent

1

2

Solve the following system of equations

2

2

3 10

Solution

Carry the augmented matrix to reduced row-echelon form

1 3

2

3 10 19

r r

r r r

x y

x y z x y z

                                        3 20

1

0

0 0 20

0

0 0

reduced row echelon matrix

leading one

x2,x4 are nonleading

variables, so we set

x2=t and x4=s

(parameters) and then

compute x1, x3

x1 = + 2t - s x2 = t x3 = + 2s x4 = s

1 1

2 0

1 2 0

1

1

2 2

0 0

0 0 0 0 0

   

   

   

  

   

      

   

  

   

   

       

   

Theorem 2

Suppose a system of m equations in n variables has a solution If the rank of the augment matrix is r then the set of solutions involves exactly n-r parameters

rankA=2

leading one

4(number of variables)- 2(rankA) =2 (two parameters : x2=t, x4=s)

1 1 3

2 0 0

1 2 0 0 0

1

1

0

     

     

     

    

     

        

1.3.Homogeneous Equations (phương trình nhất)

• The system is called homogeneous (thuần nhất) if the constant matrix has all the entry are zeros

• Note that every homogeneous system has at least one solution (0,0,…,0), called trivial solution

(nghiệm tầm thường)

• If a homogeneous system of linear equations has

Hệ mà số ẩn nhiều số pt

Theorem 1

If a homogeneous system of linear equations

has more variables than equations, then it has nontrivial solution (in fact, infinitely many)

System of equations Summary

System of Inconsistent ( no solutions)

Cosistent Unique solution

(exactly one solution)

Infinitely many solutions

linear equations yes yes yes

linear equations that has more

variables than equations

yes no yes

homogeneous linear

equations no yes yes

homogeneous linear equations that has more variables than equations

Summary

Elementary Operations Gaussian Elimination