MT3334: Advanced Linear Algebra
This is the coursepage for the advanced linear algebra course for August 2026 at IISER, Pune. In this course, we study the abstract vector spaces over arbitrary fields, diagonalization and canonical forms for linear maps. We explore the close connection between Linear algebra and geometry of Euclidean spaces R^n
and C^n via inner products and quadratic forms.
Announcements:
Please join Google classroom using this link:
https://classroom.google.com/c/ODcxMjY2MDUyNjE2?cjc=jpkidwro
There will be total 10 quizzes as part of part of continuous assessment. Four of the them will be before mid-semester exam. Six of them will be after mid-semester examination.
Tutorial will happen on every Thursday with updated timings 8:30 AM -10:00 AM at LHC 301.
If you have any doubts or want to discuss we suggest you mail one of us and we will fix a time.
Homework
Weekly homework assignments will be uploaded on the Google Classroom page for this course by Thursday evening. These do not need to be submitted and will not be graded.
Tests
A short test will be taken at the beginning of each tutorial session (20 minutes) based on the homework assignment of the previous week. We will consider best of 8 out of 10.
Exams
There will be one mid-semester exam and one final exam.
Grading
The final score is calculated as follows:
Weekly tests (40%)
Mid-semester exam (30%)
End-semester exam (30%)
The two lowest weekly-test scores will be dropped before calculation, but no further relaxations will be made for sickness, emergencies, institute events etc. No email will be answered regarding this.
Lectures
Mondays and Tuesday : 9 am -10 am in LHC 301
Tutorials
Thursday : 9 am -10 am in LHC 301
Instructor
Debargha Banerjee, Email: debargha@iiserpune.ac.in
Teaching Assistants
Aakash Gupta, Email: aakash.gupta@students.iiserpune.ac.in , office number is 454, Main academic building, IISER Pune
Akshay Kharade, Email: kharade.akshay@students.iiserpune.ac.in, office number is 458, Main academic building, IISER Pune
Textbooks:
2. Introduction to Linear Algebra: G. Strang (2009), Wellesley Cambridge Press
3. Linear Algebra done right: S. Axler (2014) Springer
4. Linear Algebra with applications: Bretscher (2012), Pearson
5. Linear Algebra: K. Hoffman and R. Kunze (2009), Prentice Hall
6. Linear Algebra: A Geometric Approach by S. Kumaresan
7. Algebra: M. Artin (1991) Prentice Hall
8. Linear Algebra: Stephen Friedberg, Arnold Insel (2004)
Week 1
3/8/26:-Fields, Vector spaces over any fields, examples, subspaces, span
4/8/26:-Linear independence of vectors, Basis, existence of bases, dimension, finite-dimensional vector spaces, Assignment 1 uploaded.
6/8/26:-Tutorial.
Week 2
10/8/26:-Sums, direct sums, dimension formula
11/8/26:-Quotient spaces, linear maps, kernel, image, rank and nullity, Assignment 2 uploaded.
13/8/26:-Tutorial.
Week 3
17/8/26:-Matrix representation of linear maps, change of basis, isomorphisms
18/8/26:-Eigenvalues and eigenvectors, characteristic polynomial
20/8/26:-Tutorial. Quiz 1
Week 4
24/8/26:-Cayley–Hamilton theorem and applications
25/8/26:-Minimal polynomial, relation with characteristic polynomial, diagonalizability
27/8/26:-Tutorial. Quiz 2
Week 5
Instructor will not be in town.
31/08/26: Problem solving on week 1 and 2 (to be taken by tutors)
01/09/26:Problem solving on week 3 and 4 (to be taken by tutors)
04/09/26: Tutorial. Quiz 3
Week 6
07/09/26: Idea of canonical forms (statement of Jordan Canonical forms), diagonalisation, triangulation
08/09/26: Trace and determinant, properties and geometric interpretations
11/09/26: Tutorial. Quiz 4.
Week 7
14/09/26: Dual spaces, dual basis, annihilators
15/09/26: Adjoint (transpose) of linear transformations, natural pairing, double dual
18/09/26: Tutorial. Quiz 5.
Mid-semester exam and break.
Week 8
05/10/26: Bilinear forms, symmetric and alternating bilinear forms
06/10/26:Quadratic forms, symmetric matrices, change of coordinates
9/10/26: Tutorial.
Week 8
12/10/26: Reduction of quadratic forms, real quadratic forms
13/10/26: Orthogonal matrices and orthogonal groups, unitary matrices and unitary groups
15/10/26: Tutorial, Quiz 6.
Week 9
19/10/26: Hermitian matrices, spectral theorem for real symmetric matrices
20 /10/26: Dusherra (Holiday)
22/10/26: Tutorial, Quiz 7.
Week 10
26/10/26:Spectral theorem for Hermitian matrices, orthogonal/unitary diagonalization
27 /10/26: Introduction to classical linear groups: GL(n), SL(n), O(n), SO(n), U(n), SU(n), Sp(2n) (basic definitions and examples)
29/10/26: Tutorial, Quiz 8.
Week 11
02/11/26:Tensor product of vector spaces: universal property, construction, examples
03/11/26:Basis and dimension of tensor products, bilinear maps and tensor products
05/10/26: Tutorial, Quiz 9.
Week 12
09/11/26:Symmetric tensors, symmetric square (S^2(V)), examples
10/11/26: Alternating tensors, exterior square (\Lambda^2(V)), decomposition (V\otimes V=S^2(V)\oplus\Lambda^2(V)) (when (\mathrm{char}(F)\neq2))
12/11/26: Tutorial, Quiz 10.