Chapter 1 Linear Algebra S 2 Systems of Linear Equations.

Chapter 1

Linear Algebra

S 2

Systems of Linear Equations

Ch1_2

Definition• ax + by= c ; a,b,c is called a ………………..

• The graph of such equation is a ………….……. in xy-plane.

• The system of two linear equations is like:

………………………………….

• If ……… satisfy the equations we called them…………...

• In this system the solution is …………

1.1 Matrices and Systems of Linear Equations

, , 0a b

Definition A linear equation in n variables x1, x2, x3, …, xn has the form

…………………………………………

where the coefficients a1, a2, a3, …, an ,b

Ch1_3

Figure 1.2……. solution–2x + y = 3–4x + 2y = 2 Lines are ………..No point of intersection. No solutions.

Solutions for system of linear equations

Figure 1.1……… solution x + 3y = 9–2x + y = –4 Lines …………….Unique solution: x = 3, y = 2.

Figure 1.3………. solutions4x – 2y = 66x – 3y = 9 Both equations have the …………………... Any point on the graph is a solution. Many solutions.

Ch1_4

The following is an example of a system of three linear equations:

How to solve a system of linear equations? For this we introduce a method called ………………………………..

62

3 32

2

321

321

321

xxx

xxx

xxx

Ch1_5

1 2 3

1 2 3

1 2 3

2

2 3 3

2 6

x x x

x x x

x x x

Relations between System of linear equations and Matrices

We use matrices to describe system of linear equations:

1. The coefficients of the variables form a matrix

called the ………………………..

2. The coefficients together with the constant terms form a matrix

called the ………………………..

Note

Ex:matrix of coefficients augmented matrix

Ch1_6

Elementary Row Operation 1. ……………. two rows of a matrix. 2. ………… the elements of a row by a nonzero

………..

3. Add a ………… of the elements of one row to the corresponding elements of another row.

Elementary Row Operations of Matrices

ij i jR R R

ikR

i jkR R

Ex: 1 1 1

2 3 1

1 1 2

3 21 2 2 22

................... ...................... ......................R RR R R

Ch1_7

Example 1Solve the following system of linear equation by Gauss-Jordan Elimination

623322

321

321

321

xxxxxxxxx

Solution

Ch1_8

Example 2Solving the following system of linear equation.

83318521242

321

321

321

xxxxxxxxx

Solution

1

2

3

2

solution 1

3

x

x

x

Ch1_9

Summary

•This method of solving the system of n linear equations in n variables is called ……………………………..

•If the system has a ……….solution then A is row equivalent to ………….

•If A In, then the system has ………….. solution.

[ : ] [...... : .......]A B i.e.,

Def. [In : X] is called the ……………………….. of [A : B].

Ch1_10

Example 3: Many SystemsSolving the following three systems of linear equation, all of which have the same matrix of coefficients.

3321

2321

1321

42

for 42

3

bxxx

bxxx

bxxx

in turn 433

,210

,11118

3

2

1

bbb

Solution

Ch1_11

1.2 Gauss-Jordan EliminationDefinitionA matrix is in reduced echelon form if

1. Any rows consisting entirely of zeros are …………………………. of the matrix.

2. The first nonzero element of each other row is …... This element is called a …………..

3. The leading 1 of each row after the first is positioned to the……… of the leading 1 of the previous row.

4. All other elements in a column that contains a leading 1 are ……..

5. The reduced echelon form of a matrix is ………..

Ch1_12

Examples for reduced echelon form

10000

04300

03021

9100

3010

7001

3100

0000

4021

000

210

801

(…..) (…..)(…..) (…..)

Ch1_13

Gauss-Jordan Elimination

System of linear equations augmented matrix reduced echelon form solution

Ch1_14

Example 1Use the method of Gauss-Jordan elimination to find reduced echelon form of the following matrix.

1211244129333

22200

Solution

Ch1_15

Example 2Solve, if possible, the system of equations

7537429333

321

321

321

xxxxxxxxx

Solution

……. sol.

Ch1_16

Example 3Solve, if possible, the system of equations

1 2 3

2 3

1 2 3

5 3

3 1

2 8 3

x x x

x x

x x x

Solution

…… sol.

Ch1_17

Homogeneous System of linear Equations

Definition A system of linear equations is said to be ……………….... if

all the constant terms are ……...

Example:

0632

052

321

321

xxx

xxx

Observe that is a solution. 1 2 3....., ....., .....x x x

Theorem 1.1

A system of homogeneous linear equations in n variables always has the solution x1 = 0, x2 = 0. …, xn = 0. This solution is called the …………………...

Ch1_18

Homogeneous System of linear Equations

Theorem 1.2

A system of homogeneous linear equations that has …………….. than …………… has ……….. solutions.On of these solutions is the trivial solution.

0632

052

321

321

xxx

xxxExample: