July-November, 2025
Lectures : Monday, Tuesday and Thursday at 2:00 PM
Office Hour : Wednesday from 2:00PM to 4:00 PM
Minor Examination - 25%
Major Examination - 45%
Quizzes - 30%
A student is expected to have full attendance in the course unless student takes leave of absence for valid medical or bonafide reasons. In any case, at least 75% attendance in the course is mandatory.
1. The course aims to develop a deep understanding of abstraction and mathematical rigor, which are essential for advanced mathematical research.
2. The course will emphasize rigorous proofs and the development of problem-solving skills in linear algebra and group theory through regular assignments and practice problem sheets.
At the end of the course the students will be able to:
1. explain the fundamental concepts of advanced algebra such as groups, group actions and their role in modern mathematics and applied contexts.
2. understand vectors, matrices, and their canonical forms.
3. learn singular value decomposition along with its applications.
1. Group ( 6 Lectures) : Recall definitions of Group, Subgroup, Normal subgroup, Simple group, quotient group. Cyclic group, Abelian group, Dihedral group, Quaternion group, Permutation group, Classical Groups such as general linear group and special linear group. Group Homomorphism and Isomorphism Theorems, Group automorphism. Introduction to computer algebra software GAP.
2. Group Action ( 12 Lectures ) : Definition and examples, conjugation action, Conjugacy classes and class equation, Orbit stabilizer lemma, Burnside’ s lemma: Proof and applications in counting techniques, Cauchy and Sylow Theorems : Proof and applications in identifying simple group. Classification of finite abelian groups, Overview of classification of finite simple groups, Direct Product and Semidirect product, Solvable and Nilpotent group.
3. Vector Space (9 Lectures) : Recall definitions of vector space, subspace, Linear independence, Basis, dimension, quotient space, linear transformation and matrices, eigenvalues, eigen vectors, characteristic and minimal polynomial, diagonalization of a matrix. Primary decomposition: Reduction to canonical forms such as triangular form, Jordan form and rational canonical form, Dual spaces.
4. Inner product space and Canonical Forms (9 Lectures) : Inner product, Orthogonality and Gram Schmidt, Direct Sum, Operators on inner product spaces, Spectral Theorems, Bilinear and quadratic forms. Classical groups.
5. Singular Value Decomposition (3 Lectures): Singular Values and Singular Vectors, Polar Decomposition, Singular Value Decomposition and its applications.
1. Dummit, D.S., & Foote, R.M., Abstract Algebra, Third Edition, Wiley
2. Blyth, T.S., & Robertson, E.F., Further Linear Algebra (2002), Springer.
3. Strang, G., Linear Algebra for Everyone, (2020), Cambridge Press.
1. Lang, S., Algebra, Revised Third Edition, Springer.
2. Lang, S., Linear Algebra, Third Edition, Springer.
3. Strang, G., Linear Algebra and learning from data, (2019), Cambridge Press.
31-July-2025 : Introduction to the Course, Definition of groups and subgroups, Abelian groups
4,5,7 -August-2025 : Normal subgroup, Simple group, quotient group. Cyclic group, Abelian group, Dihedral group, Quaternion group, Permutation group, Classical Groups such as general linear group and special linear group.
11,12,14 -August-2025 : Group Homomorphism and Isomorphism Theorems, Group automorphism. Introduction to computer algebra software GAP. Classification of finite abelian groups, Overview of classification of finite simple groups
18,19,21-August-2025 : Definition of group action and examples, conjugation action, Conjugacy classes and class equation, Orbit stabilizer lemma, Quiz-I
25,26,28-August-2025 : Burnside’ s lemma: Proof and applications in counting techniques, Cauchy’s Theorem with proof.
1,2,4 -September-2025 : Three Sylow Theorems : Proof and applications
8,9,11 -September-2025 : Sylow Theorems : applications in identifying simple group, direct product and Semi direct product
15 -September-2025 : Solvable and Nilpotent group.
17- 20-September-2025 : Minor Examination
22,23,25 -September-2025 : Recall definitions of vector space, subspace, Linear independence, Basis, dimension, quotient space, linear transformation and matrices, eigenvalues, eigen vectors, characteristic and minimal polynomial, diagonalization of a matrix.
29,30-September-2025 : Inner product, Orthogonality and Gram Schmidt, Direct Sum
6,7,9-October-2025 : Operators on inner product spaces, Spectral Theorems, Quiz-II
13,14,16-October-2025 : Primary decomposition: Reduction to triangular form
27,28,30-October-2025 : Jordan form and rational canonical form
3,4,6-November-2025 : Dual spaces, Bilinear and quadratic forms. Classical groups, Quiz-III
10,11,13-November-2025 : Singular Values and Singular Vectors, Polar Decomposition, Singular Value Decomposition and its applications.
20-26 November 2025: Major Examination
July-November, 2025
Lectures : Monday, Tuesday and Thursday at 2:00 PM
Office Hour : Wednesday from 2:00PM to 4:00 PM
Minor Examination - 25%
Major Examination - 45%
Quizzes - 30%
A student is expected to have full attendance in the course unless student takes leave of absence for valid medical or bonafide reasons. In any case, at least 75% attendance in the course is mandatory.
1. The course aims to develop a deep understanding of abstraction and mathematical rigor, which are essential for advanced mathematical research.
2. The course will emphasize rigorous proofs and the development of problem-solving skills in linear algebra and group theory through regular assignments and practice problem sheets.
At the end of the course the students will be able to:
1. explain the fundamental concepts of advanced algebra such as groups, group actions and their role in modern mathematics and applied contexts.
2. understand vectors, matrices, and their canonical forms.
3. learn singular value decomposition along with its applications.
1. Group ( 6 Lectures) : Recall definitions of Group, Subgroup, Normal subgroup, Simple group, quotient group. Cyclic group, Abelian group, Dihedral group, Quaternion group, Permutation group, Classical Groups such as general linear group and special linear group. Group Homomorphism and Isomorphism Theorems, Group automorphism. Introduction to computer algebra software GAP.
2. Group Action ( 12 Lectures ) : Definition and examples, conjugation action, Conjugacy classes and class equation, Orbit stabilizer lemma, Burnside’ s lemma: Proof and applications in counting techniques, Cauchy and Sylow Theorems : Proof and applications in identifying simple group. Classification of finite abelian groups, Overview of classification of finite simple groups, Direct Product and Semidirect product, Solvable and Nilpotent group.
3. Vector Space (9 Lectures) : Recall definitions of vector space, subspace, Linear independence, Basis, dimension, quotient space, linear transformation and matrices, eigenvalues, eigen vectors, characteristic and minimal polynomial, diagonalization of a matrix. Primary decomposition: Reduction to canonical forms such as triangular form, Jordan form and rational canonical form, Dual spaces.
4. Inner product space and Canonical Forms (9 Lectures) : Inner product, Orthogonality and Gram Schmidt, Direct Sum, Operators on inner product spaces, Spectral Theorems, Bilinear and quadratic forms. Classical groups.
5. Singular Value Decomposition (3 Lectures): Singular Values and Singular Vectors, Polar Decomposition, Singular Value Decomposition and its applications.
1. Dummit, D.S., & Foote, R.M., Abstract Algebra, Third Edition, Wiley
2. Blyth, T.S., & Robertson, E.F., Further Linear Algebra (2002), Springer.
3. Strang, G., Linear Algebra for Everyone, (2020), Cambridge Press.
1. Lang, S., Algebra, Revised Third Edition, Springer.
2. Lang, S., Linear Algebra, Third Edition, Springer.
3. Strang, G., Linear Algebra and learning from data, (2019), Cambridge Press.
31-July-2025 : Introduction to the Course, Definition of groups and subgroups, Abelian groups
4,5,7 -August-2025 : Normal subgroup, Simple group, quotient group. Cyclic group, Abelian group, Dihedral group, Quaternion group, Permutation group, Classical Groups such as general linear group and special linear group.
11,12,14 -August-2025 : Group Homomorphism and Isomorphism Theorems, Group automorphism. Introduction to computer algebra software GAP. Classification of finite abelian groups, Overview of classification of finite simple groups
18,19,21-August-2025 : Definition of group action and examples, conjugation action, Conjugacy classes and class equation, Orbit stabilizer lemma, Quiz-I
25,26,28-August-2025 : Burnside’ s lemma: Proof and applications in counting techniques, Cauchy’s Theorem with proof.
1,2,4 -September-2025 : Three Sylow Theorems : Proof and applications
8,9,11 -September-2025 : Sylow Theorems : applications in identifying simple group, direct product and Semi direct product
15 -September-2025 : Solvable and Nilpotent group.
17- 20-September-2025 : Minor Examination
22,23,25 -September-2025 : Recall definitions of vector space, subspace, Linear independence, Basis, dimension, quotient space, linear transformation and matrices, eigenvalues, eigen vectors, characteristic and minimal polynomial, diagonalization of a matrix.
29,30-September-2025 : Inner product, Orthogonality and Gram Schmidt, Direct Sum
6,7,9-October-2025 : Operators on inner product spaces, Spectral Theorems, Quiz-II
13,14,16-October-2025 : Primary decomposition: Reduction to triangular form
27,28,30-October-2025 : Jordan form and rational canonical form
3,4,6-November-2025 : Dual spaces, Bilinear and quadratic forms. Classical groups, Quiz-III
10,11,13-November-2025 : Singular Values and Singular Vectors, Polar Decomposition, Singular Value Decomposition and its applications.
20-26 November 2025: Major Examination