Course Details:
Systems of linear equations, elementary row operations, rank of a matrix, Gaussian elimination, elementary matrices, inverse of a matrix.
Determinants : Determinant as area and volume, determinant using permutations, properties of determinants.
Vector Spaces: Vector Spaces over fields, subspaces, linear independence, bases, dimension.
Linear Transformations: Linear transformations, algebra of linear transformations, Rank Nullity Theorem and applications, isomorphism, matrix representation of linear transformations, change of bases, transpose of a linear transformation.
Inner Product Spaces: Inner products, Gram-Schmidt orthogonalization, orthogonal projections and best approximation, linear functionals and adjoint operator, bilinear maps, quadratic forms, symmetric, hermitian, unitary and normal operators.
Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors, characteristic polynomial, minimal polynomials, Cayley-Hamilton Theorem, triangulation and diagonalization, Finite dimensional spectral theorem for normal operators, Singular Value Decomposition, Jordan canonical form.
Text Books:
1. Serge Lang, Linear Algebra, Springer (1987)
2. Kenneth Hoffman and Ray Kunze, Linear Algebra, Prentice Hall India Learning Private Limited; 2nd edition (1978)
3. Sheldon Axler, Linear Algebra Done Right, 2nd Edition, Springer UTM, 1997.
Reference Books:
1. Gilbert Strang, Linear Algebra and its Applications, Cengage Learning
2. Paul R. Halmos, Finite Dimensional Vector Spaces, Springer
3. S. Kumaresan, Linear Algebra: a geometric approach, Prentice Hall India
4. Serge Lang, Introduction to linear algebra, Springer
5. M. T. Nair, A. Singh, Linear Algebra, Springer (2018)
6. F. R. Gantmacher, The Theory of Matrices, Volume 1, American Mathematical Soc. (1959)
7. F. R. Gantmacher, The Theory of Matrices, Volume 2, American Mathematical Soc. (1959)