Course Details
Subject {L-T-P / C} : MA6107 : Advanced Linear Algebra { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Prof. Suvendu Ranjan Pattanaik
Syllabus
Vector spaces, Linear transformations, kernel, and image of a linear transformation, Rank-nullity theorem, Change of bases for linear transformations, Base-change Matrices, Orthonormal bases, Gram-Schmidt process. Invariant subspaces, Cayley-Hamilton Theorem, minimal polynomial, Adjoint operators, Normal, unitary, and self-adjoint operators, Schur's Lemma, Spectral theorem for normal operators, Diagonalization, Triangulation of a matrix, Direct-sum decomposition, Cyclic subspaces, and Annihilators, Rational and Jordan canonical forms, LU and Cholesky decomposition, Householder’s Reflection, Strum’s Sequence, SVD, Tridiagonal Matrix, QR and Polar Decomposition, Projection Matrix, Generalized inverse of the matrix, Perron–Frobenius theorem. Quadratic Form. Sylvester Inertia Theorem, Properties of positive and non-negative matrices, Linear functional, Inner product spaces.Surface and volume in R^n, Stability of a matrix, Lyapunov Stability theorem.
Course Objectives
- To know the application of linear algebra and matrices in the different branches of mathematics.
- To introduce theoretical aspects of linear algebra required for recent evolving branches like machine learning and data analysis.
- Also, introduces Perron–Frobenius theorem, Quadratic form. Sylvester inertia theorem, Linear functional, Bilinear mapping.
- To introduce Inner product spaces, surface and volume in R^n, Stability of a matrix, Lyapunov Stability theorem.
Course Outcomes
Students should well-versed with the application of linear algebra and matrices in the different branches of mathematics.
Essential Reading
- K. Hoffman and R. A. Kunze, , Linear Algebra, Prentice Hall of India
- H. Dym, Linear algebra in Action, American Mathematical Society
Supplementary Reading
- R A Horn, C R Johnson, Matrix Analysis, Cambridge
- J H Kwak, S Hong, Linear Algebra, Birkhauser