Time : Wed 3,4 and Fri 3,4
Credits : 4+4
Textbook :
1. Linear Algebra, 4th Edition, by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence
2. Linear Algebra in Action, Harry Dym
Many problems in science are often resolved by firstly linearizing them. For instance, in calculus, differentiation can be seen as a process of locally approximating functions with linear functions. This process of linearization often sheds new light into the original problem. Linear algebra, the branch of mathematics that systematically deals with linearized functions, is now considered the fundamental language in almost all quantitative scientific studies.
The objective of this course is to introduce the fundamental ideas of linear algebra. In the first semester, our focus will be on studying vector spaces and the linear maps defined between them. Although these concepts are defined in abstract and general terms, we will approach them in a concrete and computable manner using matrices. By the end of the first semester, we will discover that every linear map on a finite dimensional vector space can be represented by an upper triangular matrix, and in some cases, even better, by a diagonal matrix, depending on the choice of a `basis’.
This course primarily focuses on pure mathematics. Theorems will be rigorously proven in their most general form, but key concepts and results will be illustrated through concrete examples from various disciplines.
This is a second course in linear algebra. In this course, three main topics will be discussed in details. The first main topic is on Jordan canonical form and its related results. For an endomorphism T on a complex vector space of finite dimension, neither its geometric multiplicities nor its minimal polynomial is sufficient in characterising it up to conjugation by invertible matrices. The theory of Jordan canonical form resolves this problem completely and sheds new light on our understanding of linear operators that are not necessarily diagonalisable. Secondly we treat (real or complex) vector spaces endowed with an appropriately defined “inner product”. These structures are ubiquitous and fundamental in mathematics and many parts of the sciences. We will cover orthonormal basis, Gram-Schmidt process, spectrum theorems, singular value decomposition (SVD), Moore–Penrose inverses and their applications , among other things. In the final part of the course, we will discuss some contemporary topics in linear algebra. We will focus on matrices with non-negative entries. As applications, we will discuss Markov chain, stochastic matrices and prove the Perron-Frobenius theorem which leads to the mathematics behind Google.
[Semester 1]
(Part I. Vocabulary in Linear Algebra)
Week 1. Fields and Vector spaces, Subspaces
Week 2. Linear independence, Spanning, Basis and Dimension
Week 3. Direct sums and complementary subspaces
Week 4. Linear maps (I) : kernel and image, rank and nullity theorem
Week 5. Linear maps (II) : representation by matrices
Week 6. Linear maps (III) : Gaussian elimination, rank of a matrix
Week 7. Linear maps (IV) : inverse of a linear map, transition matrices
Week 8. Midterm Exam Week
(Part II. Theory of Diagonalization)
Week 9. Determinants (I) : Definitions, properties, Cramer's rule
Week 10. Determinants (II) : Multi-linearity, Block matrices
Week 11. Diagonalization (I) : Eigenvalues, eigenvectors and eigenspaces
Week 12. Diagonalization (II) : algebraic and geometric multiplicities
Week 13. Diagonalization (III) : Cayley-Hamilton, minimal polynomials
Week 14. Applications to differential and difference equations
Week 15. Dual spaces and duality theorems
Week 16. Final Exam Week
[Semester 2]
(Part III - Jordan Canonical Form)
Week 1. Jordan Canonical Form (I) : The statement, examples
Week 2. Jordan Canonical Form (II) : Jordan chains and generalized eigenspaces
Week 3. Jordan Canonical Form (III) : The proof
(Part IV - Inner Product Spaces)
Week 4. Inner product space (I) : definition and examples
Week 5. Inner product space (II) : Gram-Schmidt process
Week 6. Adjoint operators : normal, unitary, self-adjoint
Week 7. Spectral Theorem : statement and proof
Week 8. Midterm Exam Week
(Part V - Applications of Linear Algebra)
Week 9. Singular value decomposition (SVD)
Week 10. Moore-Penrose's inverse (pseudoinverse)
Week 11. Bilinear and quadratic forms
Week 12. Definiteness of matrices, Second derivative tests
Week 13. Markov chain
Week 14. Perron-Frobenius Theorem and mathematics behind Google
Week 15. Representation theory of finite groups
Week 16. Final Exam Week