MAT 660 - Linear Algebra

Syllabus for MAT 660 - Linear Algebra

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

Prerequisites: MAT 240 and 260, or an undergraduate algebra course with basics of groups and fields.

Overview: The theory of vector spaces is an indispensible tool for Mathematics, Physics, Economics and many other subjects. This course aims at providing a basic understanding and some immediate applications of the language of vector spaces and morphisms among such spaces.

Detailed Syllabus:

1. Familiarity with sets: Finite and infinite sets; cardinality; Schroeder-Bernstein Theorem; statements of various versions of Axiom of Choice.
2. Vector spaces: Fields; vector spaces; subspaces; linear independence; bases and dimension; existence of basis; direct sums; quotients.
3. Linear Transformations: Linear transformations; null spaces; matrix representations of linear transformations; composition; invertibility and isomorphisms; change of co-ordinates; dual spaces.
4. Systems of linear equations: Elementary matrix operations and systems of linear equations.
5. Determinants: Definition, existence, properties, characterization.
6. Diagonalization: Eigenvalues and eigenvectors; diagonalizability; invariant subspaces; Cayley-Hamilton Theorem.
7. Canonical Forms: The Jordan canonical form; minimal polynomial; rational canonical form.

Main References:

• Friedberg, Insel and Spence: Linear Algebra, 4^th edition, Prentice Hall India
• Hoffman and Kunze: Linear Algebra, 2^nd edition, Prentice Hall India

Other references:

• Paul Halmos: Finite Dimensional Vector Spaces, Springer India
• Sheldon Axler: Linear Algebra Done Right, 2^nd edition, Springer International Edition
• S. Kumaresan: Linear Algebra: A Geometric Approach, Prentice Hall India