Syllabus

Vector spaces, Bases and dimensions, Sums and direct sums, Quotient spaces. Linear transformations, Kernel, and image of a linear transformation, Rank-nullity theorem, Representation of linear transformations by matrices, (8 classes)

Change of bases for linear transformations, Base-change Matrices, Orthonormal bases, Gram-Schmidt process. Invariant subspaces, Cayley-Hamilton theorem, Minimal polynomial, Adjoint operators (matrix), Normal, unitary, and self-adjoint operators (matrix), Schur's Lemma, Spectral theorem for normal operators (matrix) (Unitary diagonalisation, and triangulation of a matrix), LU and Cholesky decomposition (16 classes)

Direct-sum decomposition, Cyclic subspaces, Rational and Jordan canonical forms, Householder’s Reflection, QR and Polar Decomposition, Tridiagonal Matrix, Strum’s Sequence, Projection Matrix, Singular value decomposition, Generalised inverse of the matrix, (8 classes)

Non-negative matrix, Perron–Frobenius theorem, Quadratic form. Sylvester's inertia theorem, Linear functional, Bilinear mapping and inner product spaces. (8 classes)

To introduce the application of linear algebra and matrices in the different branches of mathematics.

Introduce the theoretical aspects of linear algebra required for emerging branches such as machine learning and data analysis.

Also, it extensively introduces students to the generalised inverse, QR decomposition and SVD.

Also, introduces Perron–Frobenius theorem, Quadratic form. Sylvester's inertia theorem, Linear functional, Bilinear mapping.

Students should be well-versed in applying linear algebra and matrices across different branches of mathematics (such as machine learning and data analysis).

Students will develop programming skills to accommodate linear algebra theories.

Students should acknowledge the benefits of linear algebra in handling big data analysis and other computer-related tasks.

Through the bilinear form, students should understand the geometric structure of the quadratic form of a Hermitian matrix.

K. Hoffman and R. A. Kunze, Linear Algebra, Prentice Hall of India

H. Dym, Linear algebra in Action (Graduate studies in Mathematics, American Mathematical Society

R A Horn, C R Johnson, Matrix Analysis, Cambridge

J H Kwak, S Hong, Linear Algebra, Birkhauser