MT2123 Advanced Linear Algebra
(August 24 semester, IISER Pune)
Instructor: Dr Anupam Singh
Audience: 2nd year BS-MS students at IISER Pune
Schedule: Lecture Wednesday, Thursday 11 AM (LHC 103)
Tutorial Friday 9 AM (LHC 103, 303, 304, 305).
Tutors: Adithya Pillai, Apurva Pujari, Adrija Dalal, Sahil Joshi, Aravind R
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:
Vector spaces over a field, basis and dimension, linear maps, adjoints of linear transformations and dual spaces.
Eigenvectors and eigenvalues, Cayley-Hamilton theorem, minimal polynomial, characteristic polynomials, trace, determinant, Jordan decomposition, Jordan canonical forms.
Quadratic forms and symmetric matrices, orthogonal and unitary matrices, diagonalization of Hermitian and symmetric matrices, introduction to classical groups.
Multilinear algebra: tensor product of two vector spaces and, decomposition of V into symmetric and alternating tensors.
Textbooks and reference material :
Linear Algebra: K. Hoffman and R. Kunze
Basic + Advanced Linear Algebra: Blyth and Robertson
Introduction to Linear Algebra: Strang
Advanced Linear Algebra: Roman
Finite Dimensional Vector Spaces: P. Halmos.
Linear Algebra done right: S. Axler
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 :
1/8/24: Introduction and motivation
2/8/24: Fields, Example of fields, vector space over a field.
7/8/24: Example of vector spaces, basic properties.
8/8/24: Subspaces, intersection and sum of subspaces.
9/8/24: Tutorial
14/8/24: basis and dimension
15/8/24: Holiday
16/8/24: Tutorial
21/8/24: Existence of basis, uniqueness of size of bases
22/8/24: extension of l.i. set to a basis, linear transformation
23/8/24: Tutorial
24/8/24: (extra class) Matrix of a Linear Transformation, Formula for the dimension of a sum of two subspaces.
28/8/24: Rank-Nullity Theorem, Dual vector space
29/8/24: Adjoint of a linear transformation, Hyperplanes
30/8/24: Quiz
4/09/24: Change of basis, similarity of matrices
5/09/24: examples, why similarity classes?
6/09/24: Tutorial
11/09/24: Idea of canonical forms (statement of Jordan Canonical forms), diagonalisation, triangulation
12/09/24: Characteristic polynomial, minimal polynomial, eigen values and eigenvectors
13/09/24: Quiz
18/09/24: Tutorial
Mid sem exam 23/09/24
03/10/24: Polynomials, End(V) a ring.
04/10/24: Tutorial
09/10/24: Application topic: PageRank Algorithm, Markov Chain, Perron-Frobenius Theorem
10/10/24: Direct sum (internal, external) of vector spaces, T-invariant subspaces
11/10/24: Tutorial
16/10/24: Example of T-invariant subspaces
17/10/24: Roots of the minimal and characteristic polynomial, Distinct eigenspaces
18/10/24: Tutorial
23/10/24: T-conductor, Diagonalisation
24/10/24: Triangulation, Cayley-Hamilton Theorem, Application of Jordan Canonical Form
25/10/24: Quiz
30/10/24: No class
31/10/24: Holiday (Deepavali)
01/11/24: Tutorial
06/11/24: Symmetric bilinear forms, classification over real and complex
07/11/24: Orthogonal group
08/11/24: Tutorial
09/11/24: Quiz (11 AM)
13/11/24: Inner product space, Hermitian form, Unitary group
14/11/24: SAGEMath demo, the group SU(2) and quaternions
15/11/24: Holiday
20/11/24: No class
End Sem exam - 21/11/24 at 10 AM
Repeat Exam 01/01/24 at 10 AM (please check the official communications from the Dean's office).