Post on 11-Jan-2016
(eq3) 23972
(eq2) 20783
(eq1) 4 2
zyx
zyx
zyxExample
Sec 3.1 Introduction to Linear System
Sec 3.2 Matrices and Gaussian Elemination
n) (eq aa aa
2) (eq aa aa
1) (eq aa aa
1nn3n32n21n1
12n323222121
11n313212111
bxxxx
bxxxx
bxxxx
n
n
n
equationsLinear of
System
)( 19 )( 523
eq2
eq1
yxyx
The graph is a line in xy-plane
The graph is a line in xyz-plane
(eq3) 23972
(eq2) 20783
(eq1) 4 2
zyx
zyx
zyxExample
Sec 3.1 Introduction to Linear System
972
783
121
23972
20783
4121
Coefficient Matrix3 x 3
Augmented Coefficient Matrix3 x 4
Sec 3.2 Matrices and Gaussian Elemination
23972
20783
4121Example
Sec 3.1 Introduction to Linear System
Augmented Coefficient Matrix3 x 4
Sec 3.2 Matrices and Gaussian Elemination
Size, shape
row
column
columnth -j and rowth -iin element denotes ija
23aExample 7
n) (eq aa aa
2) (eq aa aa
1) (eq aa aa
1nn3n32n21n1
12n323222121
11n313212111
bxxxx
bxxxx
bxxxx
n
n
n
System
Linear
nnnn
n
n
aaa
aaa
aaa
21
22221
11211
nnnnn
n
n
baaa
baaa
baaa
21
222221
111211
Coefficient Matrixn x n
Augmented Coefficient Matrixn x (n+1)
Sec 3.1 Introduction to Linear System
Sec 3.2 Matrices and Gaussian Elemination
Example
Sec 3.1 Introduction to Linear System
Sec 3.2 Matrices and Gaussian Elemination
199 123
yxyx
systemlinear theofsolution a is 2
1
y
x
Three Possibilities
Linear System
Unique Solution
1
02
yxyx
Example
Infinitely many
solutions
2
2221
yxyx
Example
NoSolution
3
21
yxyx
Example
InconsistentInconsisten
tconsistent
consistent
How to solve any linear system
Triangular system
*
*
*
z
y
x
Use back substitution
****
****
****
****
****
****
zyx
zyx
zyx Augmented
*100
**10
***1
Elementary Row OperationsMultiply one row by a nonzero constant1 iR * C
Interchange two rows
2ji R R
Add a constant multiple of one row to another row
3ji R R * C
(eq3) 11892
(eq2) 5 23
(eq1) 221083
zyx
zyx
zyxExample
Triangular system
****
****
****
*100
**10
***1
How to solve any linear system
****
****
****
****
****
***1
***0
***0
***1
***0
**10
***1
**00
**10
***1
*100
**10
***1
(eq3) 23972
(eq2) 20783
(eq1) 4 2
zyx
zyx
zyxExample
(-3) R1 + R2
(-2) R1 + R3
(-3) R2 + R3
Augmented Matrix
23972
20783
4121
15730
8420
4121(1/2) R2
3100
4210
4121
15730
4210
4121
3100
4210
4121
Convert into triangular matrix
triangular matrix
Example Convert into triangular matrix
(eq3) 11892
(eq2) 5 23
(eq1) 221083
zyx
zyx
zyx
How to solve any linear system
Triangular system
*
*
*
z
y
x
Use back substitution
****
****
****
****
****
****
zyx
zyx
zyxAugmented
*100
**10
***1
(eq3) 23972
(eq2) 20783
(eq1) 4 2
zyx
zyx
zyx
ExampleSolve
3100
4210
4121
(eq3) 23972
(eq2) 20783
(eq1) 4 2
zyx
zyx
zyxExample
(-3) R1 + R2
(-2) R1 + R3
(-3) R2 + R3
Augmented Matrix
23972
20783
4121
15730
8420
4121(1/2) R2
3100
4210
4121
15730
4210
4121
Definition: (Row-Equivalent Matrices)
A and B are row equivalent if B can be obtained from A by a finite sequence of elementary row operations
A
B
Convert into triangular matrix
Example A and B are row equivalent
Example
23972
20783
4121
3100
4210
4121
Definition: (Row-Equivalent Matrices)
A and B are row equivalent if B can be obtained from A by a finite sequence of elementary row operations
A B
A and B are row equivalent
A is the augmented matrix of sys(1)B is the augmented matrix of sys(2)
The
orem
1:
A and B are row equivalent&
sys(1) and sys(2) have same solution
Echelon Matrix
000000
430120
000000
020000
110101
zero row
Example How many zero rows
000000
000000
100000
021020
110101
010000
000001
100000
021020
110101
Echelon Matrix
non-zero row
Example
1) How many non-zero rows
2) Find all leading entries
000000
000000
100000
021020
110101
010000
000001
100000
021020
110101
000000
430120
000000
020000
110101
leading entry The first (from left) nonzero element in each nonzero row
0000
1120
1101
Echelon Matrix
Def: A matrix A in row-echelon form if
1) All zero rows are at the bottom of the matrix
2) In consecutive nonzero rows the leading in the lower row appears to the right of the leading in the higher row
1 5 0 2
0 1 0 1
0 0 0 0
A
1
1 5 0 2
0 2 0 1
0 0 0 0
A
2
1 5 0 2
0 0 1 1
0 1 0 1
A
3
1 5 0 2
0 0 0 0
0 1 0 1
A
1100000
0120000
1010000
2020010
000
000
000
4A
111
111
111
5A
How to transform a matrix into echelon form
****
****
**** Echelon
Gaussian Elimination
111010
001110
010100
1101001) Locate the first nonzero column
2) In this column, make the top entry nonzero
3) Use this nonzero entry to (below zeros )
4) Repeat (1-3) for the lower right matrix
111010
110100
010100
001110
31 RR
41 RR
110100
110100
010100
001110
Example
Echelon Matrix
Reduce the augmented matrix to echelon form.
age160Example5/p
275610637103842
10232
54321
54321
54321
xxxxxxxxxx
xxxxx
741000
320100
1012321
How to solve any linear system
Gaussian Elimination
*
*
*
z
y
x
Use back substitution
****
****
****
****
****
****
zyx
zyx
zyx Augmented
Echelon
Leading variables and Free variables
0100000
9103100
4011211
leading Variables
1x 2x 3x 4x 5x 6x
Free Variables
631 ,, xxx542 ,, xxx
AlgorithmBack Substitution
1) Set each free variable to parameter ( s, t, …)
2) Solve for the leading variables. Start from last row.
5710
10251
2Ex4/page16
x y z
sz Second row gives:
zy 75 sy 75
first row gives:zyx 2510
ssx 2)75(510 sx 3335
Thus the system has an infinite solution set consisting of all (x,y,z) given in terms of the parameter s as follows
sx 3335 sy 75
sz
23
Example
2100
6310
5211
Back Substitution
The linear systems are in echelon form, solve each by back substitution
Ex1/p162
051000
0731310
0117251
Ex10/p162
24
Quiz #1 on Saturday
Sec 3.1 + Sec 3.2