線性代數
LINEAR ALGEBRA
2005 Spring
教師及助教資料
 教師：李程輝
 Office : ED 828 ext. 31563
 助教：葉易霖
林怡文
 Lab: ED 823 ext. 54570
參 考 資 料
 Textbook:
S.J. Leon, Linear Algebra
with Applications, 6th Ed., Prentice Hall,
2002.
 Reference: R. Larson and B.H.
Edwards, Elementary Linear Algebra,
4th Ed., Houghton Mifflin Company,
2000.
成績算法
 Homework
(10 %)
 Middle Exam x2 (60 % each)
 Final Exam
(30 %)
-------------------------Total：100 %
Several Applications

How many solutions do Ax  b have?
It may have none, one or infinitely many
solutions depending on rank(A) and whether
b  col ( A) or not.
 How
to solve the following Lyapunov and
Riccati equations:
AX  XA  Q  0
A X  XA  XBR B X  Q  0
T
1
Matrix Theory
T
 Find
the local extrema of f :     C
n
2
f  0 & definiteness of the Hessian matrix.
 How
to determine the definiteness of a real
symmetric matrix?
eigenvalues
 How
to determine the volume of a
parallelogram?
Determinant
 How
to find the solutions or characterize the
dynamical behaviors of a linear ordinary
differential equation?
Eigenvalues, Eigenvectors
vector space and linear independency
 How
to predict the asymptotic( Steady-state)
behavior of a discrete dynamical system ( p280.)
Eigenvalues & Eigenvectors
 xi 
 Given  y , i  1, , n
 i
Find the best line y  C0  C1 x
i.e., find C0 & C1 
n
y
i 1
i
 (C0  C1 xi )
to fit the data.
2
is minimum
Least Square problem
(Orthogonal projection)
 How
to expand a periodic function as sum of
different harmonics? ( Fourier series)
Orthogonal projection
 How
to approximate a matrix by as few as data?
Digital Image Processing
Singular Value Decomposition
 How
to transform a dynamical system into one
which is as simple as possible?
Diagnolization, eigenvalues and eigenvectors
 How
to transform a dynamical system into a
specific form ( e.g., controllable canonical
form)
Change of basis
課程簡介
 Introduction
to Linear Algebra
 Matrices and Systems of Equations





Systems of Linear Equations
Row Echelon Form
Matrix Algebra
Elementary matrices
Partitioned Matrices
 Determinants
 The Determinants of a Matrix
 Properties of Determinants
 Cramer’s Rule
 Vector






Spaces
Definition and Examples
Subspace
Linear Independence
Basis and Dimension
Change of Basis
Row Space and Column Space
 Linear
Transformations
 Definition and Examples
 Matrix Representations of Linear
Transformations
 Similarity
 Orthogonality





The Scalar Product of Euclidean Space
Orthogonal Subspace
Least Square Problems
Inner Product Space
Orthonormal Set
 Eigenvalues





Eigenvalues and Eigenvectors
Systems of Linear Differential Equations
Diagonalization
Hermitian Matrices
The Singular Value Decomposition
 Quadratic Forms
 Positive Definite Matrices
Exercise for Chapter 1
 P.11:
 P.28:
 P.62:
 P.76:
 P.87:
 P.97:
9,10
8,9,10
12,13,21,*22,23,27
3(a,c),*6,12,18,23
*18
Chapter test 1-10
Exercise for Chapter 2
 P.105:
1,*11
 P.112: 5-8,*10-12
 P.119: 2(a,c),4,7,*8,11,12
 P.123: Chapter test 1-10
Exercise for Chapter 3
 P.131:
 P.142:
 P.154:
 P.161:
 P.173:
 P.180:
 P.186:
3-6,13,15
1,*3,5-9,13,14,16-20
5,*7-11,14-17
3,*5,7,9,11,13,15,16
1,4,*7,10,11
3,6-9,12,*13,16,17,19-21
Chapter test 1-10
Exercise for Chapter 4
 P.195:
1,8,*9,12,16,18-20,23,24
 P.208: *3,5,11,13,18
 P.217: *5,7,8,10-15
 P.221: Chapter test 1-10
Exercise for Chapter 5
 P237:
6,7,10,*13,14.
 P247: 2,9,11,*13,14,16.
 P258: *5,7,9,10,12
 P267: 4,8,9,26,*27,28,29
 P286: 2,4,*12~14,16,19,22,23,25,33
 P297: 3~5,12,*4
 P310: Chapter test 1~10
Exercise for Chapter 6
 P323
 P351
 P363
 P380
 P395
 P403
 P421
: 2~16 , 18 , *19 , 22~*25 , 27
: 1 (a)*(e) , 4 , 6 , 7 , 9~12 , 16~18 ,
23(b) , 24(a) , 25~28
: 8 , 10~*13 , *19 , 21
: *5 , 6
: 3(a)(b) , 7(a)(b) , 8~14 , *12
: *3 , 8~13
: Chapter test 1~10