linearalgebra
Vector Space, Rings and Modules MTH320
(January - April 2015)
Instructor : Dr. Anupam Singh
Schedule : Lectures on Monday and Tuesday at 11:30
Tutorials on Thursday at 13:30
Evaluation : Test I 25%
mid sem exam 25%
Test II / mini-projects 25%
end sem exam 25%
Prerequisite : None
Goal of the course :
To understand linear algebra better and see its application. We hope to be able to visualize, at least familiarize ourselves, more than 3 dimensional spaces.
Proposed Course Content :
Vector Spaces, Matrices and linear transformations. Diagonalisation and upper triangulation. Bilinear Forms, Orthogonal and Symplectic groups.
Rings, homomorphisms, ideals and quotient rings. Chinese remainder theorem, field of fractions of an integral domain.
Modules, homomorphisms, Structure theory of finitely generated modules over PID.
Jordan and rational canonical forms of matrices, Revisit Linear groups.
Textbooks and references :
Linear Algebra : Hoffman and Kunze
Further Linear Algebra: Blyth and Robertson
Abstract Algebra: Dummit and Foote
Advanced Linear Algebra : Roman
Topics for mini projects :
Classification of finite simple groups - Simplicity of PSL(n,k)
Simplicity of PSp(2l,k)
Cartan-Dieudonne Theorem
Introduction to Linear Algebraic Groups
Introduction to Lie Algebras
Mathematics behind Robotics
Using Quaternions in graphics - Gimble lock problem
Parabolic subgroups of GL(n,k)
Irreducible representations of finite Abelian groups
Tensor product and Determinant using exterior power
Application in Mathbiology/Operations Research
Jordan Decomposition and application to computing exponential
Using BLENDER to design 3D animations
C-programme to compute Bruhat Decomposition
Programme to compute Jordan Canonical form
Topological groups - Matrix Groups
Perron-Frobenius theorem
Linear algebra and internet search engines
Lorentz group
5 January - Introduction, definition of field and vector space
6 January - examples
8 January - Linear dependence and linear independence
12 January - Basis and dimension
13 January - Existence of basis and coordinates
15 January - Tutorial - All exercises of Chapter 2 and 3 from Hoffman and Kunze
19 January - Linear Transformation, matrix of a linear transformation
20 January - Rank-Nullity Theorem, Dual Vector space
22 January - Change of basis, Similar matrices
26 January - Holiday
27 January - Tutorial
29 January - Tutorial
30 January - Test - I at 10AM.
02 February - Dual Vector Space, Hyperplanes, Solving a system of Linear equations (geometric view point)
03 February - Double Dual, Transpose of a linear map
05 February - Tutorial
09 February - Eigen values, eigen vectors, similarity classes / conjugacy classes
10 February - characteristic and minimal polynomials
12 February - Tutorial
16 February - Rings, properties of Polynomial Ring k[X]
17 February - Holiday
19 February - Tutorial
Mid Sem Exam 24 February at 3PM Mid-sem-exam
02 March - Cayley-Hamilton theorem
03 March - diagonalisation and triangulation
05 March - No Class
09 March - Block-diagonal decomposition
10 March - Quotient space and Triangulation
12 March - Tutorial
16 March - Classification of Nilpotent Matrices
17 March - Jordan Canonical Forms
19 March - Tutorial
23 March - Rings with identity, ideal, homomorphism
24 March - Quotient ring, example of Z and k[X]
26 March - Class rescheduled for 27 March due to India-Australia world cup match
27 March - Tutorial
30 March - Modules over a ring
31 March - Structure theory of Module over PID
02 April - Mini project presentations - day long
03 April - Mini project presentations - whole day
06 April - No lecture due to Foundation day celebrations
07 April - Bilinear forms, symmetric and symmetric forms, orthogonal and symplectic groups
09 April - Classification of symmetric and skew-symmetric bilinear forms
13 April - Sesqui-linear forms and unitary groups
14 April - Test II
16 April - Quaternions and Orthogonal groups
18 April - Tutorials
24 April - End-sem exam