MT2123 Advanced Linear Algebra

(August 24 semester, IISER Pune)

Instructor: Dr Anupam Singh

Audience: 2nd year BS-MS students at IISER Pune

Schedule: TBA

Evaluation : Quizzes - 4 x 10 = 40%,   mid-sem - 30%,   end-sem - 30%

Prerequisite: Linear Algebra

Goal of the course: This course goes into depth on the subject of linear algebra. While the basic course on linear algebra mostly focuses on solving equations using matrices, here we study the abstract vector spaces over arbitrary fields. The subject of linear algebra is fundamental to Mathematics, Physics, Data Sciences and any computation via modelling. In this course, we learn diagonalization and canonical forms for linear maps. We explore the close connection between Linear algebra and the geometry of Euclidean spaces.

Proposed Content: 

1. Vector spaces over a field, basis and dimension, linear maps, adjoints of linear transformations and dual spaces. 

2. Eigenvectors and eigenvalues, Cayley-Hamilton theorem, minimal polynomial, characteristic polynomials, trace, determinant, Jordan decomposition, Jordan canonical forms.

3. Quadratic forms and symmetric matrices, orthogonal and unitary matrices, diagonalization of Hermitian and symmetric matrices, introduction to classical groups. 

4. Multilinear algebra: tensor product of two vector spaces and, decomposition of V into symmetric and alternating tensors.

Textbooks and reference material :

1. Linear Algebra: K. Hoffman and R. Kunze 

2. Finite Dimensional Vector Spaces: P. Halmos.

3. Linear Algebra done right: S. Axler 

4. Basic + Advanced Linear Algebra: Blyth and Robertson

5. Introduction to Linear Algebra: Strang

6. Advanced Linear Algebra: Roman

Related courses taught in the past: MTH 390   January-April 2009      Vector Space, Rings and Modules  MTH320     MT 1223 Linear Algebra

Are you bored? Try these exercises!  From Linear Algebra to Algebraic Groups

Weekly topics covered :