Linear Algebra II

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Linear Algebra I (MAT 160)

Overview: In MAT 260 we studied real and complex linear transformations up to the diagonalizability of symmetric operators. In this course we take up vector spaces over arbitrary fields and more advanced results on expressing linear transformations by simple matrices.

Detailed Syllabus:

1. Linear Equations – Systems of linear equations, matrices, elementary row operations and row reduction.
2. Vector spaces – Abstract vector spaces, subspaces, dimension, coordinates.
3. Linear transformations – Matrix representations, change of basis, linear functionals and the double dual, transpose.
4. Determinants – Commutative rings, determinant function, permutations, properties.
5. Canonical Forms – Characteristic values, invariant subspaces, simultaneous diagonalization and triangulation, invariant direct sums, Primary Decomposition Theorem, cyclic subspaces, Rational Form, Jordan Form.
6. Inner Product Spaces – Linear functionals and adjoints, unitary and normal operators, spectral theory.

References:

1. Kenneth Hoffman and Ray Kunze, Linear Algebra, 2^nd edition, PHI Learning.
2. Friedberg, Insel and Spence, Linear Algebra, 4th edition, PHI Learning
3. Sheldon Axler, Linear Algebra Done Right, 2^nd edition, Springer International Edition
4. Paul Halmos, Finite Dimensional Vector Spaces, 2^nd edition, Springer International Edition
5. Paul Halmos, Linear Algebra Problem Book, Mathematical Association of America, 1995.

Past Instructors: Neha Gupta