HCM City University of Technology – VNUHCM
Faculty : Applied Sciences
Department: Applied Mathematics
HCMCity, date 9. Mar. 2016
Course Description
ADVANCED LINEAR ALGEBRA (ALA)
Code :
- Credits
: 4 (3.1.4)
- Contact Hour - Total: 56 Lecture: 28 Tutorial lab
- Major
- Assessment:
- Pre-Requisistes
- Co-Requisistes
- Note
:
:
:
14
Optional seminars: 14
+ Computational and Applied Mathematics
+ Optimization and Modeling
Score 1 :
20% Midterm exam (writing) (30-45')
Score 2:
20% Seminar work (speaking): (60-90')
Score 3:
60% Final exam (writing) (90-120')
Linear algebra (A2)
Differential equations (A4)
Summary:
This subject is designed for the Statistical and Applied Mathematical Sciences program at HCMUT.
The aim of this subject is to present selective topics of Computational Mathematics, especially for
optimization and modeling of complex networks, dynamic systems and stochastic processes. The
lectures introduce major techniques of linear differential equations and numerical linear algebra to
be employed in Mathematical Modeling and Simulation of industrial and biological phenomena.
The learner is then able to learn more and/or conduct small research assignments themselves in
special subjects of mathematical modeling, reliability engineering, various dynamic systems in
biology, epidemiology and in economics.
References:
[1] Morris W. Hisch, Stephen Smale, UC Berkeley, Differential Equations, Dynamical Systems
and Linear Algebra, 1980
[2] Nguyen Van Minh Man, HCMUT, Computer-Algebraic Methods for the Construction of
Designs of Experiments, Ph.D. thesis, TUe, 2005
[3] Martin Anthony and Norman Biggs, LSE, Mathematics for Economics and Finance,
Cambridge University Press, 1996
[4] Jim Hefferon, Saint Michael's College, Vermont, USA, Linear Algebra, 2001
[5] Nguyen Van Minh Man, HCMUT, Symbolic Computation I and II, 2006
[6] Arjeh M. Cohen, Eindhoven University of Technology, Some Tapas of Computer Algebra, 2000
Instructor:
Nguyễn Văn Minh Mẫn, Ph.D. Schedule:
Week
1,2
Contents
Part I: Motivated topics of ALA
Chapter 1: Introductory specific topics
0. Input-Output Analysis
1. Dimensional Analysis: A basic tool in simulation
References
Note
[3],[4]
Lecture and
Reading
PĐT, Sample 2005-ĐC
Course Description: 533581549
Week
Contents
2. Line of Best Fit: a practical demand in statistical
inference
3. Markov Chains: a multipurpose tool
4. Computing Softwares: OpenModelica 1.4
References
Note
Chapter 2: Algorithms of Eigenvalue Problems
0. Origins of Eigenvalue Problems
Stability of Dynamic Systems
Bifurcation Analysis
Macro-economics-- Leontiev's linear model of
production
1. Review of basic facts
2. Polynomials of Maps and Matrices
3. Nilpotent matrices
4. Jordan Canonical Form
5. Method of Powers
6. Stable populations (Optional seminar 1)
Chapter 3: Linear differential equations for dynamic
systems
0. Review
 Vector spaces-- Basis-- Dimension
 Linear mappings-- Matrix-- Linear equations
1. Introductory linear differential systems (LDS)
 Motivation of studying X' = AX
 Options of matrix A
2. LDSs-- A is diagonalizable over R
 Examples
 Characteristic polynomial of a linear operator
 Diagonalizable linear operators
 Criteria for diagonalizing a linear operator
 Application
3. LDSs-- A has both real and complex eigenvalues
 Introductory examples
 Complex vector spaces
 Linear operator having complex eigenvalues
 Application
4. LDS of any square matrix A
 Exponential function of linear operators
5. Application in non-homogeneous LDSs (Optional
seminar 2)
[4]
Lecture and
Reading
[1]
Lecture and
Reading
11,12, Chapter 4: Multiplication maps and matrices
0. Introductory multiplication maps
 Ideals of a polynomial ideal
 Groebner basis of a polynomial ideal
1. Monomial basis of univariate polynomials (n=1)
2. Monomial basis of multivariate polynomials (n>1)
3. Multiplication maps and their matrices if n = 1
4. Multiplication maps and their matrices if n > 1
5. Eigenvalues & eigenvectors of multiplication
[2]
Lecture and
Reading
Part II: Advanced Linear Algebra Methods
3,4,
5,6
7,8,
9,10
PĐT, Sample 2005-ĐC
Course Description: 533581549
Week
Contents
References
Note
matrices
Part III: New applications of ALA
13
Chapter 5: PageRanking on web
1. Introductory Page-Ranking
2. Formulation
3. Solution
[4]
Reading and
Seminar
14
Chapter 6: Use of Multiplication matrices in reliability
engineering
1. Finiteness Criterion
2. Border basis
3. Constructing a balanced fractional factorial design
4. Could we employ designs in industrial
manufacturing? (Optional seminar 3)
Review
Appendix:
1. The method of Groebner basis
2. Basic theory of balanced fractional factorial designs
[2]
Reading and
Seminar
[5], [6]
Reading
15
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