Group and Ring Theory and Linear Algebra (B.Sc Mathematics)

Advanced group theory, rings, ideals, vector spaces, linear transformations and inner product spaces.

6 महीनेअवधि 6Chapters30टॉपिक

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सिलेबस · पाठ्यक्रम

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01Unit 1: Advanced Group Theory 5 टॉपिक

Automorphisms and inner automorphisms
Conjugacy classes and class equation
Cauchy's theorem
Sylow theorems (statements and applications)
Direct products

02Unit 2: Rings and Subrings 5 टॉपिक

Definition and examples of rings
Integral domains and fields
Zero divisors and units
Characteristic of a ring
Subrings

03Unit 3: Ideals and Ring Homomorphisms 5 टॉपिक

Ideals and quotient rings
Prime and maximal ideals
Ring homomorphisms and isomorphism theorems
Polynomial rings
Euclidean domains, PID and UFD (introduction)

04Unit 4: Vector Spaces 5 टॉपिक

Vector spaces and subspaces
Linear span and linear independence
Basis and dimension
Sum and direct sum
Quotient space

05Unit 5: Linear Transformations 6 टॉपिक

Linear transformations
Kernel and range
Rank-nullity theorem
Matrix of a linear transformation
Change of basis
Eigenvalues and diagonalisation

06Unit 6: Inner Product Spaces 4 टॉपिक

Inner product and norm
Cauchy-Schwarz inequality
Orthogonality
Gram-Schmidt orthogonalisation

परिचय

आप यह कर पाएँगे

Use conjugacy, the class equation and Sylow theorems in finite groups
Identify rings, integral domains and fields and their characteristic
Work with ideals, quotient rings and ring homomorphisms
Find bases and dimensions of vector spaces and subspaces
Find kernel, range and matrices of linear transformations and apply rank-nullity
Use inner products and the Gram-Schmidt process

यह किसके लिए है

B.Sc Mathematics students of Semester 5 (UP State Universities, NEP 2020)
Students preparing for end-semester theory papers of Mathematics

आपको क्या चाहिए

Algebra of Semester 3 (groups and homomorphisms)
Matrices of Semester 2

सवाल

Which syllabus does this subject follow?

It follows the B.Sc Mathematics Semester 5 syllabus of the UP State Universities (NEP 2020 common minimum syllabus). Where your own university's syllabus differs, the centre follows your university syllabus.

How are students assessed at the centre?

Through unit-wise multiple-choice tests, a final test paper and doubt-clearing sessions that follow the university examination pattern.

Will this help in the semester examination?

Yes. Units are taught in the university order with solved numericals and exam-pattern questions.