本課程以英語授課。數學系學生修課不計入畢業學分。輔數學系的學生得替代線性代數二。

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, spectral 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. The syllabus can be found below. (Part III - Jordan Canonical Form) Week 1. Jordan Canonical Form (I) : The statement, examples and applications 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 (Part V - Applications of Linear Algebra) Week 9. Singular value decomposition (SVD) Week 10. Moore-Penrose's inverse 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

After finishing this course, students are expected to 1. be able to compute the Jordan canonical form of a given endomorphism over a complex vector space; 2. be familiar with finite-dimensional inner product spaces to a very general extent, both computationally and theoretically 3. be able to state and prove Spectral Theorems for normal and symmetric operators of an inner product space; 4. have a working knowledge of Markov chain and stochastic matrices and understand the mechanism behind the page rank algorithm of Google.

Students are expected to have taken a first course in linear algebra, such as MATH4018 (Intro. to Linear Algebra 1) or linear algebra courses offered by other departments. To be specific, we expect students to have known a reasonable set of vocabulary and fundamental results in linear algebra. For example, - Vector space over a field - Linear maps between vector spaces and their correspondence with matrices - Kernel and image of a linear map, Rank and nullity theorem - Change of coordinate matrices - Eigenvalues, eigenvectors and eigenspaces - Determine whether a given square matrix (over C) is diagonalizable

Besides the 4-hour lectures per week, students should expect to spend around 2-3 hours weekly in digesting the lecture materials as well as completing exercises offered by the lecturer or the teaching assistant(s).

We will be using materials from 1. S. H. Friedberg, A. J. Insel, L. E. Spence, "Linear Algebra", 4th Edition, Pearson Education, 2014 ; ISBN, 0321998898. 2. H. Dym, "Linear Algebra in Action"

These books would also be useful, for example, 1. K. Hoffman and R. Kunze, "Linear Algebra". 2. Herstein and Winter, "Matrix theory and linear algebra".