Curriculum and Syllabi - Indian Institute of Technology Bhubaneswar

Detail SyllabusSemester-ITotal Credits = 221. Linear Algebra (MA4001):3-0-0: 3 Credits Prerequisite:NilVector spaces over fields, subspaces, bases and dimension; Systems of linear equations, matrices,rank, Gaussian elimination; Linear transformations, representation of linear transformations bymatrices, rank-nullity theorem, duality and transpose; Determinants, Laplace expansions, cofactors,adjoint, Cramer's Rule; Eigenvalues and eigenvectors, characteristic polynomials, minimalpolynomials, Cayley-Hamilton Theorem, triangulation, diagonalization, rational canonical form,Jordan canonical form; Inner product spaces, Gram-Schmidt orthonormalization, orthogonalprojections, linear functionals and adjoints, Hermitian, self-adjoint, unitary and normal operators,Spectral Theorem for normal operators; Rayleigh quotient, Min-Max Principle. Bilinear forms,symmetric and skew-symmetric bilinear forms, real quadratic forms, Sylvester's law of inertia,positive definiteness.Texts:1. S. Axler, Linear Algebra Done Right, 2nd edition, UTM, Springer, 1997.2. M. Artin, Algebra, Prentice Hall of India,1994.References:1.M. Artin, Algebra, Prentice Hall of India,1994.2.K. Hoffman and R. Kunze, Linear Algebra, Pearson Education (India), 2003. Prentice-Hall of India, 1991.3.S. Lang, Linear Algebra, Undergraduate Texts in Mathematics, Springer-Verlag, NewYork, 1989.4.H.E. Rose, Linear Algebra, Birkhauser, 2002.5.G. Strang, Linear Algebra and its applications. 4 th edition2. Real Analysis (MA4002):3-1-0: 4 Credits Prerequisite:NilReal number system and set theory: Completeness property, Archimedian property, Denseness ofrationals and irrationals, Countable and uncountable, Cardinality, Zorn’s lemma, Axiom of choice.Metric spaces: Open sets, Closed sets, Continuous functions, Completeness, Cantor intersectiontheorem, Baire category theorem, Compactness, Totally boundedness, Finite intersection property.Rlemann-Stieitjes integral: Definition and existence of the integral, Properties of the integral,Differentiation and integration. Sequence and Series of functions: Uniform convergence, Uniformconvergence and continuity, Uniform convergence and integration, Uniform convergence anddifferentiation. Equicontinuity, Ascoli’s Theorem, Weierstrass approximation theorem.Texts:1. T. Apostol, Mathematical Analysis, 2nd ed., Narosa Publishers, 2002.2. W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1983.References:1. E. Hewitt and K. Stomberg, Real and Abstract Analysis: A Modern Treatment of the Theory of Functionsof a Real Variable, Springer, 1975.