資訊科學數學11 :
Linear Equation and Matrices
陳光琦助理教授 (Kuang-Chi Chen)
chichen6@mail.tcu.edu.tw
1
Linear Algebra
Content of B. Kolman and D. R. Hill, Linear Algebra,
8th edition
Chap. 1 Linear Equations and Matrices
• Linear systems
• Matrices
• Dot Product and Matrix Multiplication
• Properties of Matrix Operations
• Matrix Transformations
• Solutions of Linear Systems of Equations
• The Inverse of a Matrix
• LU-Factorization (Optional)
2
Linear Algebra
Chap. 3 Determinants
 Definition and Properties
 Cofactor Expansion and Applications
 Determinants from a Computational Point of View
Chap. 4 Vectors in Rn
 Vectors in Plane
 n-Vectors
 Linear Transformations
3
Linear Algebra
Chap. 6 Real Vector Spaces
 Vector Spaces
 Subspaces
 Linear Independence
 Basis and Dimension
 Homogeneous Systems
 The Rank of a Matrix and Applications
 Coordinates and Change of Basis
 Orthonormal Bases in Rn
 Orthogonal Complements
4
Linear Algebra
Chap. 8 Eigenvalues, Eigenvectors, and
Diagonalization
 Eigenvalues and Eigenvalues
 Diagonalization
 Diagonalization of Symmetric Matrices
5
Linear Algebra
Chap. 2 Applications of Linear Equations and
Matrices (Optional)
 An Introduction to Coding
 Graph Theory
 Computer Graphics
 Electrical Circuits
 Markov Chains
 Linear Economic Models
 Introduction to Wavelets
6
Linear Algebra
Chap. 5 Applications of Vectors in R2 and R3
(Optional)
 Cross Product in R3
 Lines and Planes
Chap. 7 Applications of Real Vector Spaces (Optional)
 QR-Factorization
 Least Squares
 More on Coding
7
Linear Algebra
Chap. 9 Applications of Eigenvalues and Eigenvectors
(Optional)
 The Fibonacci Sequence
 Differential Equations (Calculus Required)
 Dynamical Systems (Calculus Required)
 Quadratic Forms
 Conic Sections
 Quadric Surfaces
8
Linear Algebra
Chap. 10 Linear Transformations and Matrices
Definition and Examples
 The Kernel and Range of a Linear Transformation
 The Matrix of a Linear Transformation
 Introduction to Fractals (Optional)
Chap. 11 Linear Programming (Optional)
 The Linear Programming Problem: Geometric
Solution
 The Simplex Method
 Duality
 The Theory of Games
9
Linear Algebra
Chap. 12 MATLAB for Linear Algebra
 Input and Output in MATLAB
 Matrix Operations in MATLAB
 Matrix Powers and Some Special Matrices
 Elementary Row Operations in MATLAB
 Matrix Inverse in MATLAB
 Vectors in MATLAB
 Applications of Linear Combinations in MATLAB
 Linear Transformations in MATLAB
 MATLAB Command Summary
10
Linear Algebra
Appendix A Complex Numbers
A.1 Complex Numbers
A.2 Complex Numbers in Linear Algebra
Appendix B Further Directions
B.1 Inner Product Spaces (Calculus Required)
B.2 Composite and Invertible Linear Transformations
11
Linear Equations and Matrices
Linear Systems
12
Linear Systems
1.1 Linear systems
• What is a linear equation?
ax  b
• Variables and constants
a1 x1  a2 x2    an xn  b
• Unknowns and solutions
13
A Solution to A Linear Equation
• A solution to a linear equation
6 x1  3 x 2  4 x3  13
62   33  4 4   13
14
A Linear System
• A linear system
a11 x1  a12 x 2    a1n x n  b1
a 21 x1  a 22 x 2    a 2 n x n  b2






a m1 x1  a m 2 x 2    a mn x n  bm
• A system of m linear equations in n unknowns
15
A Common Method
• A commonly used
method to find
solutions to a linear
system is the method
of elimination
• Example 1
x  y  100, 000
0.05 x  0.09 y  7800

x  y  100, 000
0.04 y  2800

y  70, 000
x  30, 000
16
Elimination Method: Example 2
• Example 2 – no solution
x  3 y  7
2x  6 y  7

x  3 y  7
0 x  0 y  21
Contradiction !
17
Example 3 – A Unique Solution
• Example 3 – a unique solution
x  2 y  3z  6
x  2 y  3z  6
x  2 y  3z  6
2 x  3 y  2 z  14   7 y  4 z  2   7 y  4 z  2
 5 y  10 z  20
3 x  y  z  2
y  2z  4
x  2 y  3z  6
y  2z  4
 7 y  4z  2

x  2 y  3z  6
x  2 y  3z  6
y  2z  4
10 z  30
y  2z  4
z 3

18
An Over-Determined Example
• Example 4 – an over-determined linear system
that has many solutions
x  2 y  3z  4
2 x  y  3z  4

x  2 y  3z  4
 3 y  3z  12

y  z4
x  4  2 y  3 z
 4  2  z  4   3 z
 z4

xr4
y r4
zr
3 variables,
2 equations.
possible solutions :
x  5, y  3, z  1
x  2, y  6, z  2
x and y are lead variables, and z is a free variable.
19
An Under-Determined Example
• Example 5 – an under-determined linear
system that has a unique solution
x  2 y  10
x  2 y  10
x  2 y  10

 y4
2 x  2 y  4
 6 y  24
y4
3 x  5 y  26
 y  4
2 variables,
3 equations.
20
Another Under-Determined Example
• Example 6 - an under-determined linear
system that has no solution
x  2 y  10
x  2 y  10
x  2 y  10
2 x  2 y  4   6 y  24  y  4
 y  10
y  10
3 x  5 y  20
Contradiction !
21
Linear Equations and Matrices
• A linear system may have one solution (a
unique solution), no solution, or infinitely
many solutions.
22
Three Elementary Operations
Three elementary operations
• Interchange two equations (E1)
• Multiply an equation by a nonzero constant
(E2)
• Add a multiple of one equation to another (E3)
23
Equivalent Systems
Equivalent systems
 Linear systems having the same solution set

The method of elimination via the three
elementary operations yields another
equivalent linear system
24
Example 7
• Example 7 - Production Planning
20  x3
x1 
2 x1  3x 2  4 x3  80
2

2 x1  2 x 2  3x3  60
x2  20  x3
x3 : 0  x3  20, x3  R
25
Matrices
26
Matrices
1.2 Matrices
• An mn matrix A
 a11 a12
a
 21 a22

A
 ai1 ai 2


 am1 am 2
a1 j
a2 j
aij
amj
a1n 

a2 n 


ain 


amn 
27
Rows & Columns
• The i-th row of A
Ai   ai1 ai 2  ain 
• The j-th column of A
• Elements of A
 a1 j 
a 
2j

Aj 
 
 
 amj 
 aij  of A
28
Example
• Example 1
 1 2 3
A


1
0
1


1 1 0
D  2 0 1
3  1 2
1 4 
B

2  3
E  3
1


C   1
 2 
F   1 0 2
29
n-Vectors
• n-vector
• Example 2
u  1 2  1 0
1


v   1
 3 
• The set of n-vectors: Rn
30
Tabular Display
• Example 3- Tabular Display of Data
Taipei
Taichung
Tainan
Hualien
 0 210 380 250 


Taichung 210
0
170
460


Tainan  380 170
0 630 


Hualien  250 460 630
0 
Taipei
31
Tabular Display
• Example 4 - Tabular Display of Production
Product 1 Product 2
Plant 1 560

Plant 2 360

Plant 3 380

Plant 4  0
340
450
420
80
Product 3
280 

270 
210 

380 
32
Display Linear Equations by Matrix
• Example 5
x  2 y  10
2 x  2 y  4
3 x  5 y  26

Ax  b
1 2 
10 
 x




A   2 2 , x    , b   4
y

 3 5 
 26 
A is the coefficient matrix
33
A Diagonal Matrix
Definition- The diagonal matrix
A square matrix A defined as follows is called the
diagonal matrix
A = [ aij ], aij = 0 , for all i≠ j .
• Example 6
-3 0 0 
4 0 
 0 -2 0 
G
,
H




0

2


 0 0 4 
34
A Scalar Matrix
Definition- The scalar matrix
A diagonal matrix A defined as follows is called the
scalar matrix
A = [ aij ], aij = c , for all i = j ,
aij = 0 , for all i≠ j .
• Example 7
1 0 0 
-2 0 


I 3  0 1 0  , J  

0
-2


0 0 1 
35
Equal Matrices
Definition- Equal matrices
Two mn matrices A = [aij] and B = [bij] are
said to be equal if
aij = bij , 1≤ i ≤m , 1≤ j ≤n .
• Example 8
1 2 1
 1 2 w




A   2 3 4  and B   2 x 4 
0 4 5 
 y 4 z 
A, B are equal if x = -3 , y = 0 , z = 5 , w = -1
36
Matrix Addition
Matrix addition
• Example 9
1 2 4
0 2 4
A
and B= 


2

1
3
1
3
1




1  0 2  2 4  (4)  1 0 0 
A B  



2

1

1

3
3

1
3
2
4

 

37
Application of Matrix Addition
• Example 10
Manufacturing Shipping
Cost
Cost
 20
F1  30
 40
Manufacturing Shipping
Cost
Cost
70
15 Model A

10  Model B F2  80
90
5  Model C
55 Model A
45 Model B
65 Model C
Compute F1 + F2
38
Scalar Multiplication
• Scalar multiplication
A = [aij], B = [bij],
bij = caij , for 1≤ i ≤m , 1≤ j ≤n .
• Difference
•
The difference of A and B : A – B
Example 11
 2 3 5
 2 1 3 
A
4 2
and B = 

1
3
5
2
 2  2 3  1 5  3 0 4 8
A B  



4

3
2

5
1

2
1

3
3

 

39
Example
• Example 12
p  18.95
14.75 8.60
0.20 p   (0.20)18.95 (0.20)14.75 (0.20)8.60
 3.79 2.95 1.72
p  0.20 p  18.95 14.75 8.60  3.79 2.95 1.72
 15.16 11.80 6.88
p  0.20 p  0.80 p
40
Linear Combination
• Linear combination of k matrices: A1, …, Ak
c1 A1 + c2 A2 + … + ck Ak .
• Coefficients: c1 , c2 , … , ck .
41
Example of Linear Combination
• Example 13 – B is a linear combination of A1 and A2
0 3 5 
 5 2 3
A1   2 3 4  and A2   6 2 3
1 2 3
 1 2 3
• B = 3A1 – 0.5A2
• Coefficients: c1 = 3 , c2 = -0.5 .
42
Example
(cont’d)
0 3 5 
 5 2 3
1


B  3  2 3 4    6 2 3
2
1 2 3
 1 2 3
 5
 2

 3

 7

 2
27 
10
2 

21 
8
2 
21 
5 
2 
43
More Examples
• Example 14-1
23  2  35 0  4 2 5
• Example 14-2
1
 0.1
0.5  4  0.4  4 
 6
0.2
44
Transpose of A Matrix
• The transpose of a matrix
A   aij 
AT   aijT 
a  a ji
for 1≤ i’ ≤n , 1≤ j’ ≤m .
T
ij
Here, A = [aij] is a mn matrix and
AT = [aji] is a nm matrix .
45
Examples of Transpose
• Example 15
6 2 4 
5 4
4

2
3


 3 1 2  , C    3 2  ,
A
,
B






 0 5 2 
0 4 3 
 2 3
2
D  3 5 1 , E   1
 3 
4 0
 6 3 0
 5 3 2 




T
T
T
A   2 5  , B   2 1 4  , C  
,
4
2

3


 3 2 
 4 2 3 
3
DT   5 , and E T   2 1 3
 1 
46