Linear Algebra
NEP Semester I (P2S01NCMTH03)
Syllabus (NEP2020)
| Unit-I | Vector Spaces: Quick review of vector spaces, examples of vector spaces including sequence spaces and function spaces, Subspaces, Bases, Dimensions, Row and column spaces, Rank and nullity, Bases for subspaces, Invertibility, Applications: The Wronskian. |
| Unit-II | Linear Transformations: Basic properties of linear transformations, Invertible linear transformations, Matrices of linear transformations, Vector spaces of linear transformations, Change of bases, Similarity, Applications: Dual spaces, Adjoint (only definition and examples). |
| Unit-III |
Inner Product Spaces: Dot products and inner products, the lengths and angles of vectors, Matrix representations of inner products, Gram-Schmidt orthogonalization, Projections, Orthogonal projections (only definitions and examples). Diagonalization: Eigenvalues and eigenvectors, Diagonalization of matrices, Exponential matrices, Diagonalization of linear transformations. |
| Unit-IV |
Jordan Canonical Forms: Basic properties of Jordan canonical forms, Generalized eigenvectors, the power $A^k$ and the exponential $e^A$, Cayley-Hamilton theorem, the minimal polynomial of a matrix. Quadratic Forms: Basic properties of quadratic forms, Diagonalization of quadratic forms, A classification of level surfaces. |
Reference Books
- Kwak J. H., Hong S., Linear Algebra, (2nd edition), Birkhauser, 2004.
- Hoffman K., Kunze R., Linear Algebra, (2nd edition), Prentice Hall of India, 2003.
- Herstein I. N., Topics in algebra, Wiley Eastern Ltd., New Delhi, (2nd Edition), 1975.
- Sahai V., Bist V., Linear Algebra, Narosa Publishing House, 2002.
- Ramachandra Rao A. and Bhimasankaram P., Linear Algebra (2nd Edition), Hindustan Book Agency, TRIM-19, 2000.
- Axler S., Linear Algebra Done Right (2nd Edition), Springer, 1997.
- Strang G., Introduction to Linear Algebra (6th Edition), Wellesley Cambridge Press, 2023.