B.Sc. Mathematics
5th Semester Syllabus
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MJC-08 : Ring Theory and Linear Algebra-I
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Unit 1: Rings and Ideals
◦Definition and examples of rings
◦Properties of rings
◦Definition and examples of Subrings
◦Zero divisors
◦Integral domains and its examples
◦Properties of integral domains
◦Division rings
◦Fields
◦Characteristic of a ring
◦Ideals and its properties
◦Quotient rings
Unit 2: Ring Homomorphisms
◦Ring homomorphisms
◦Kernel of Ring homomorphisms
◦Properties of ring homomorphisms
◦Isomorphism theorems for Rings
Unit 3: Vector Spaces
◦Vector spaces
◦Subspaces
◦Algebra of subspaces
◦Linear combination of vectors
◦Linear span
◦Linear independence
◦Basis and dimension
◦Dimension of subspaces
◦Quotient spaces
Unit 4: Linear Transformations
◦Linear transformations
◦Null Spaces and Ranges
◦Matrix representation of a linear transformation
◦Rank-Nullity theorem
◦Algebra of linear transformation
◦Eigenvalues and Eigenvectors
◦Characteristic equation of a matrix and Cayley-Hamilton theorem
Unit 5: Isomorphisms
◦Isomorphisms for vector spaces
◦Isomorphism theorems for vector spaces
◦Invertibility and Isomorphisms
Reference Books:
- Gallian, Joseph. A. (2013). Contemporary Abstract Algebra (8th ed.). Cengage Learning India Private Limited. Delhi. Fourth impression, 2015
- Herstein, I. N. (2006). Topics in Algebra (2nd ed.). Wiley Student Edition. India
- Friedberg, Stephen H., Insel, Arnold J., & Spence, Lawrence E. (2003). Linear Algebra (4th ed.). Prentice-Hall of India Pvt. Ltd. New Delhi
- Kumaresan, S. (2000). Linear Algebra: A Geometric Approach, Prentice Hall India Learning Private Limited, New title Edition
- Hoffman, Kenneth & Kunze, Ray Alden (1978). Linear Algebra (2nd ed.). Prentice-Hall of India Pvt. Limited. Delhi. Pearson Education India Reprint, 2015