B.Sc. Mathematics
7th Semester Syllabus
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MJC-13 : Ring Theory and Linear Algebra-II
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Unit 1: Polynomial Rings
◦Field of quotients of an integral domain
◦Embedding of rings
◦Polynomial rings
◦The Division algorithm and consequences
◦The Remainder Theorem
◦The Factor Theorem
◦Irreducible Polynomials
◦Reducible polynomials
◦Primitive Polynomial
◦Gauss's Lemma
◦Irreducibility tests
◦Unique factorization domains
Unit 2: Dual Spaces
◦Linear Functionals
◦Dual spaces
◦Dual basis
◦Double dual
◦Annihilators
◦Transpose of a linear transformation and its matrix in the dual basis
Unit 3: Diagonalization
◦Eigenspaces of a linear operator
◦Diagonalization of Linear Operators
◦Invariant subspaces
◦The minimal polynomial for a linear operator
Unit 4: Inner Product Spaces
◦Inner products and Norms
◦Orthonormal basis
◦Gram-Schmidt orthogonalization process
◦Orthogonal complements
◦Bessel's inequality
Unit 5: Operators on Inner Product Spaces
◦The adjoint of a linear operator
◦Least squares approximation
◦Minimal solutions to systems of linear equations
◦Normal and Self-Adjoint Operators
◦Unitary and orthogonal operators
Reference Books:
- Gallian, Joseph. A. (2019). Contemporary Abstract Algebra (9th ed.). Cengage Learning India Private Limited
- Friedberg, Stephen H., Insel, Arnold J., & Spence, Lawrence E. (2022). Linear Algebra (5th ed.), Pearson Education
- Herstein, I. N. (2006). Topics in Algebra (2nd ed.). Wiley Student Edition. India
- Hoffman, Kenneth M., & Kunze, Ray Alden (1978). Linear Algebra (2nd ed.). Prentice-Hall of India Pvt. Limited. Delhi. Pearson Education India Reprint, 2015
- Kumaresan, S. (2000). Linear Algebra: A Geometric Approach, Prentice Hall India Learning Private Limited, New title Edition