Coopération Mathématique Interuniversitaire Cambodge France,
Master Training Program
Phnom Penh, September 27 - October 20, 2010
From September 27 to October 19, 2010:
Master of Science in Mathematics, Royal University of Phnom Penh (URPP - Université Royale de Phnom Penh).
CIMPA Master Training Program.
1. LINEAR EQUATIONS
Introduction. Elementary Row Operations. Row Echelon Form. Reduced Row Echelon Form. Solving a System of Linear Equations. Homogeneous Systems. Application to Network Flow. Application to Electrical Networks. Application to Economics. Application to Chemistry. Application to Mechanics.
Introduction. Systems of Linear Equations. Inversion of Matrices. Application to Matrix Multiplication. Finding Inverses by Elementary Row Operations. Criteria for Invertibility. Consequences of Invertibility. Application to Economics. Matrix Transformation on the Plane. Application to Computer Graphics. Complexity of a Non-Homogeneous System. Matrix Factorization. Application to Games of Strategy.
Introduction. Determinants for Squares Matrices of Higher Order. Some Simple Observations. Elementary Row Operations. Further Properties of Determinants. Application to Curves and Surfaces. Some Useful Formulas. Further Discussion.
Introduction. Vectors in 2-Space. Vectors in 3-Space. Vector Products. Scalar Triple Products. Application to Geometry in 3-Space. Application to Mechanics.
5. INTRODUCTION TO VECTOR SPACES
Real Vector Spaces. Subspaces. Linear Combination. Linear Independence. Basis and Dimension.
6. VECTOR SPACES ASSOCIATED WITH MATRICES
Introduction. Row Spaces. Column Spaces. Rank of a Matrix. Nullspaces. Solution of Non-Homogeneous Systems.
7. EIGENVALUES AND EIGENVECTORS
Introduction. The Diagonalization Problem. Some Remarks. An Application to Genetics.
8. LINEAR TRANSFORMATIONS
Euclidean Linear Transformations. Linear Operators on the Plane. Elementary Properties of Euclidean Linear Transformations. General Linear Transformations. Change of Basis. Kernel and Range. Inverse Linear Transformations. Matrices of General Linear Transformations. Change of Basis. Eigenvalues and Eigenvectors.
9. REAL INNER PRODUCT SPACES
Euclidean Inner Products. Real Inner Products. Angles and Orthogonality. Orthogonal and Orthonormal Bases. Orthogonal Projections.
10. ORTHOGONAL MATRICES
Introduction. Eigenvalues and Eigenvectors. Orthonormal Diagonalization.
11. APPLICATIONS OF REAL INNER PRODUCT SPACES
Least Squares Approximation. Quadratic Forms. Real Fourier Series.
12. COMPLEX VECTOR SPACES
Complex Inner Products. Unitary Matrices. Unitary Diagonalization.
WWL Chen, Linear Algebra
Jim Hefferon, Linear Algebra
Edwin H. Connell, Elements of Abstract and Linear Algebra.
Keith Matthews, Elementary Linear Algebra
Alex Postnikov, Linear Algebra.
First assignment, October 1, 2010.
Second assignment, October 11, 2010.
Final exam, October 20, 2010.
Introduction to real analysis:
1. The Real Numbers
The Real Number System. Mathematical Induction. The Real Line.
2 Diﬀerential Calculus of Functions of One Variable
Functions and Limits. Continuity. Diﬀerentiable Functions of One Variable. L’Hospital’s Rule. Taylor’s Theorem.
3. Integral Calculus of Functions of One Variable
Definition of the Integral. Existence of the Integral. Properties of the Integral. Improper Integrals. A More Advanced Look at the Existence of the Proper Riemann Integral
4 Infinite Sequences and Series
Sequences of Real Numbers. Earlier Topics Revisited With Sequences. Infinite Series of Constants. Sequences and Series of Functions. Power Series.
5. Real-Valued Functions of Several Variables
Structure of R^n. Continuous Real-Valued Function of n Variable. Partial Derivatives and the Diﬀerential. The Chain Rule and Taylor’s Theorem.
6. Vector-Valued Functions of Several Variables
Linear Transformations and Matrices. Continuity and Differentiability of Transformations. The Inverse Function Theorem. The Implicit Function Theorem.
7. Integrals of Functions of Several Variables
Definition and Existence of the Multiple Integral. Iterated Integrals and Multiple Integrals. Change of Variables in Multiple Integrals.
8 Metric Spaces
Introduction to Metric Spaces. Compact Sets in a Metric Space. Continuous Functions on Metric Spaces
William Trench, Introduction to real analysis.