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.
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.
Announcements:
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.
Lectures
Mondays and Tuesday : 9 am -10 am in LHC 301
Tutorials
Thursday : 9 am -10 am in LHC 301
Instructors
Debargha Banerjee, Email: debargha@iiserpune.ac.in
Teaching Assistants
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
Vector spaces over a field, examples, subspaces, span, linear independence
Basis, existence of bases, dimension, finite-dimensional vector spaces
Week 2
Sum, direct sum, quotient spaces, dimension formula
Linear maps, kernel, image, rank and nullity
Week 3
Matrix representation of linear maps, change of basis, isomorphisms
Dual spaces, dual basis, annihilators
Week 4
Adjoint (transpose) of linear transformations, natural pairing, double dual
Eigenvalues and eigenvectors, characteristic polynomial
Week 5
Cayley–Hamilton theorem and applications
Minimal polynomial, relation with characteristic polynomial, diagonalizability
Week 6
Trace and determinant, properties and geometric interpretations
Review and problem-solving session (or first mid-semester test)
Mahatma Gandhi Jayanti: 02/10/2025 (Thursday)
Week 7
Bilinear forms, symmetric and alternating bilinear forms
Quadratic forms, symmetric matrices, change of coordinates
Week 8
Reduction of quadratic forms, real quadratic forms
Orthogonal matrices and orthogonal groups, unitary matrices and unitary groups
Week 9
Hermitian matrices, spectral theorem for real symmetric matrices
Spectral theorem for Hermitian matrices, orthogonal/unitary diagonalization
Week 10
Introduction to classical linear groups: GL(n), SL(n), O(n), SO(n), U(n), SU(n), Sp(2n) (basic definitions and examples)
Tensor product of vector spaces: universal property, construction, examples
Week 11
Basis and dimension of tensor products, bilinear maps and tensor products
Symmetric tensors, symmetric square (S^2(V)), examples
Week 12
Alternating tensors, exterior square (\Lambda^2(V)), decomposition (V\otimes V=S^2(V)\oplus\Lambda^2(V)) (when (\mathrm{char}(F)\neq2))
Review, advanced examples, applications, and comprehensive revision