Math Dept, Faculty of Applied Science, HCM University of Technology -------------------------------------------------------------------------------------

Math Dept, Faculty of Applied Science, HCM University of

Technology------------------------------------------------------------------------------

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Chapter 3: Systems of Linear Equations

• Instructor Dr. Dang Van Vinh (11/2006)

CONTENTS---------------------------------------------------------------------------------------------------

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I – Systems of Linear

Equations

II – Homogeneous

system

I. Systems of Linear Equations ---------------------------------------------------------------------------------------------------

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The quantities a11, a12, …, amn are called the coefficients of the system.

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

m m mn m m

a x a x a x b

a x a x a x b

a x a x a x b

A System of m linear equations in n unknowns has the form:

Definition of a System of Linear equations.

The quantities b1, b2, …, bm are called the free or constant terms of the systems.


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The solution of system is such a collection n of the numbers c1, c2, …, cm which, being substituted into system for the unknowns x1, x2, …, xn, turns all the equations of the system into identities.

A system is called homogeneous if all its constant terms b1, b2, …, bm are equal to zero.

Definition of Homogeneous System.

A system is called nonhomogeneous if at least one of the constant terms b1, b2, …, bm is different from zero.

Definition of Nonhomogeneous System.

I. Systems of Linear Equations ---------------------------------------------------------------------------------------------------------------------------

consistent system

inconsistent system

A system of linear equations has either:

1. no solution,

2. exactly one solution, or

3. infinitely many solutions

Two linear systems are equivalent, if they have the same solution set.

The basic strategy is to replace one system with an equivalent system that is easier to solve.


There are three elementary reduction operations .

An operation is called elementary reduction operation if it transforms one system to equivalent system.

Definition of elementary reduction operation

1. Swapping: an equation is swapped with another.

2. Rescaling (multiplying by a scalar): an equation has both sides multiplied by a nonzero constant.

3. Pivoting: an equation is replaced by the sum of itself and a multiple of another.

Remark: We need to prove that above three operations are elementary reduction operations.


1 2

1 3

2r rr r

0

3 3 3

3 3

x y

y z

y z

2 3r r

0

3 3 3

4 0

x y

y z

z

Solution: x = 1; y = -1; z = 0

Solve the system:

0

2 3 3

2 3

x y

x y z

x y z

Example


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1 1 0

2 1 3

1 2 1

coefficient matrix:

augmented matrix:

1 1 0 0

2 1 3 3

1 2 1 3


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1 2

1 3

2r rr r

2 3r r

1 1 0 0

2 1 3 3

1 2 1 3

1 1 0 0

0 3 3 3

0 3 1 3

1 1 0 0

0 3 3 3

0 0 4 0

I. System of Linear Equations

The variable corresponding to pivot columns in the matrix are called basic variable.

The other variable is called free variable.

Definition of Basic and Free variable.

1 1 1 2 1

2 2 3 5 6

3 3 4 1 1

row operations

1 1 1 2 1

0 0 1 1 4

0 0 0 6 8

x1, x3, x4: basic variables

x2: free variables

If , then the system AX = b is consistent. ( | ) ( )r A b r A

If , then the system AX = b is inconsistent. ( | ) ( )r A b r A

I. System of Linear Equations

If = number of unknowns , then the system AX = b has unique solution. ( | ) ( )r A b r A

If < number of unknowns , then the system AX = b has many solutions. ( | ) ( )r A b r A

The Kronecker Capelli Theorem

If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.

I. System of Linear equations------------------------------------------------------------------------------------

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1. Write the augmented matrix of the system.

2. Use the row reduction algorithm to obtain an equivalent augmented matrix is echelon form. Decide whether the system is consistent.

3. Write the system of equations corresponding to the matrix

4. Rewrite each nonzero equation from step 3 so that its one basic variable is expressed in terms of any free variable appearing in the equation.

Using Row Reduction to solve a linear system


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The augmented matrix of a linear system has been transformed by a row operations into the form below. Determine if the system is consistent.

1 5 2 6

. 0 4 7 2 ,

0 0 5 0

a

1 1 1 3

. 0 1 2 4 ,

0 0 0 5

b

1 1 1 0

. 0 1 2 5 ,

0 0 0 0

c

1 1 1 0

. 0 3 1 0 .

0 0 0 0

c

Example


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Example

5 2 1

4 6

3 3 9

x y z

x y z

x y z

Solve the following system:


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3

3 5 9 2

2 3 3

y z

x y z

x y z

Example

Solve the following system:

I. Systems of Linear Equation---------------------------------------------------------------------------------------------------

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Basis variable: 521 ,, xxx Free variable: 43 , xx

General Solution:

1

2

3

4

5

24 2 3

7 2 2

4

x

x

x

x

x

Find the general solution of the linear system

Example

2 3 4 5

1 2 3 4 5

1 2 3 4 5

3 6 6 4 5

3 7 8 5 8 9

3 9 12 9 6 15

x x x x

x x x x x

x x x x x

I. Systems of Linear Equations-----------------------------------------------------------------------------------------------------

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Find the general solutions of the systems whose augmented matrix is given as below

Example

1 1 1 1

2 3 4 1

3 4 2 1

I. Systems of Linear Equations--------------------------------------------------------------------------------------

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Example

1 1 2 0

2 1 5 0

3 4 5 0

I. Systems of Linear Equations--------------------------------------------------------------------------------------

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Example

1 1 1 1 2

2 1 3 0 1

3 4 2 2 5

2 3 1 1 3

I. Systems of Linear Equations---------------------------------------------------------------------------------------------------

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1 1 2 0 1

2 3 1 2 4

3 4 5 1 3

1 2 3 1 0

Example

I. Systems of Linear Equations------------------------------------------------------------------------------------------

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Determine the value(s) m such that the matrix is the augmented of a consistent linear system

Example

2

1 1 1

1 1 ,

1 1

m

m m

m m


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1 1 1 1

2 3 1 4

3 4 1m m

Determine the value(s) m such that the matrix is the augmented of a consistent linear system

Example


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Example Determine the value(s) m such that the corresponding

linear system has unique solution

1 1 1 1 1

2 1 3 1 2,

3 4 2 0 6

2 1 0 1m m


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Example Determine the value(s) m such that the corresponding

linear system has unique solution

2

2 3 1 4 0

3 2 1 5 7

1 1 1m m

II. Homogeneous Systems. ---------------------------------------------------------------------------------------------------

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A system is called homogeneous if all its constant terms b1, b2, …, bm are equal to zero.

Definition of Homogeneous System.

The homogeneous system always has zero solution

x1 = x2 = … = xn = 0. This solution is called the trivial solution.

The homogeneous system always possesses a unique solution – the trivial solution - if and only if rank (A) = n.


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The homogeneous system AX = 0 has nontrivial solution if and only if rank (A) < n.

The homogeneous system AX = 0, where A is a square matrix, has nontrivial solution if and only if det(A) = 0.


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Determine the general solution for the following homogeneous system.

Example

1 2 3 4

1 2 3 4

1 2 3 4

2 2 0

2 4 3 0

3 6 4 0

x x x x

x x x x

x x x x


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Among all solutions that satisfy the homogeneous system

Example

2 0

2 4 0

2 0

x y z

x y z

x y z

Determine those also satisfy the nonlinear constraint y – xy = 2z


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If A is the coefficient matrix for a homogeneous system consisting of four equations in eight unknowns and if there are five free variables, what is rank (A)?

Example

Explain why a homogeneous system of m equations in n unknowns where m < n must always possess an infinite number of solutions.

Example


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Determine the value(s) m such that the homogeneous system has nontrivial solution

Example

0

2 3 5 0

3 ( 1) 0

x y z

x y z

x my m z

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Math Dept, Faculty of Applied Science, HCM University of Technology -------------------------------------------------------------------------------------