programme

Algebra I

Home/ Algebra I
Course TypeCourse CodeNo. Of Credits
Foundation CoreSUS1MA5034

Semester and Year Offered: Monsoon Semester 2012-13

Course Coordinator and Team: Balchand Prajapati, Geetha Venkataraman

Email of course coordinator: balchand@aud.ac.in

Pre-requisites: Pre-requisite for this course is Mathematics at the XII grade level and 4 credits of Introduction to Mathematical Thinking (SUS1MA501)

Aim: This course helps students develop their ability to think abstractly within the setting offered through abstract and linear algebra. Such skills would add to a student’s ability to analyse and solve problems even in real life.

Course Outcomes:

After completing this course, students will be able

  1. to understand knowledge of complex numbers, roots of unity
  2. basic knowledge of groups, rings
  3. basic knowledge of linear algebra.

Brief description of modules/ Main modules:

The following topics will be covered in the course under the four main modules as described below.

  • Classical Algebra

Complex numbers, geometric representation of a complex number and the trigonometric form of a complex number, addition and multiplication of two ( or more) complex numbers, inverse of a non-zero complex number, their geometric interpretation and trigonometric form, De Moivre’s Theorem, nth roots of a complex number, theory of equations.

  • Group Theory

Sets and relations, equivalence relation, functions, bijective functions, composite of two or more functions, and inverse of a function: existence and uniqueness, binary operations, identity and inverse with respect to a given binary operation, uniqueness of identity and inverses, congruence relations, introduction and motivation to groups through dihedral groups, definition and examples of groups (including cartesian product of two groups), elementary properties of groups, finite groups and subgroups, subgroup tests, examples of subgroups (including cyclic subgroups, center of a group, centralizer and normalizer of subgroups, subgroup generated by subset).

  • Ring Theory

Introduction to rings, motivation and definition, examples and properties of rings, uniqueness of identity and inverses when they exist, definition of sub rings, sub ring test, examples, unity of a sub ring if it exists, definition of a zero divisor and an integral domain, examples of integral domain, cancellation law with respect to multiplication, definition of a field, finite integral domains, the field of integers modulo p, p a prime.

  • Linear Algebra

Definition of a vector space over R, examples, subspaces, linear combinations, subspace spanned by a set, null space and column space of a matrix, linear transformations, kernel and range of a linear transformation, linearly dependent and independent sets, the spanning set theorem, bases for null space and column space, coordinate systems, graphical interpretation of coordinates, coordinate mapping.

Assessment Details with weights:

S.No

Assessment

Date/period in which Assessment will take place

Weightage

1

Class test

End August

10%

2

Mid Semester Exam

End September/ early October

25%

3

Lab/ Home Assignments

Throughout the semester

15%

4

Presentation/ Viva

End October/ early November

15%

5

End Semester Exam

As per AUD Academic Calendar

35%

 

Reading List:

  • Paul E. Bland, The Basics of Abstract Algebra, W. H. Freeman and Company, 2002.
  • Joseph A. Gallian, Contemporary Abstract Algebra (Fourth Edition), Narosa Publishing House, New Delhi, 1999.
  • David C. Lay, Linear Algebra and its Applications (Third Edition), Pearson Education Asia, Indian Reprint, 2007.

Additional References:

  • Bhattacharya, Jain and Nagpal, Basic Abstract Algebra (Second Edition), Cambridge, 2009.
  • Peter J. Cameron, Introduction to Algebra (Second Edition), Oxford University Press, 2008.
  • Neal H.Mc Coy, Introduction to Modern Algebra (Fifth Revised Edition), Brown (William C.) Co, U.S., 1992.
  • John R. Durbin, Modern Algebra, An Introduction (Fifth Edition), John Wiley and Sons (Asia) Pte. Ltd, 2005.
  • Jimmie Gilbert and Linda Gilbert, Linear Algebra and Matrix Theory (Second Edition), Brooks Cole, 2004.