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Overview

The main purpose of this course in the study of linear operators on finite dimensional vector spaces. The idea is to emphasize the simple geometric notions common to many parts of mathematics and its applications. Except for an occasional reference to undergraduate mathematics, the course will be self-contained. The algebraic co-ordinate free methods will be adopted through out the course. These methods are elegant and as elementary as the classical as coordinatized treatment. The scalar field will be arbitrary (even a finite field), however, in the treatment of vector spaces with inner products, special attention will be given to the real and complex cases. Determinants via the theory of multilinear forms. Variety of examples of the important concepts. The exercises will constitute significant asstion ; ranging from routine applications to ones which will extend the very best students.

Syllabus

Week 1:  Vector Spaces
Week 2:  Linear Maps
Week 3:  Quotiet Spaces ; Affine and Projective spaces
Week 4:  Matrices
Week 5:  Determinants
Week 6:  More Determinants
Week 7:  Linear Operators
Week 8:  Diagonalizable and Triangulable operators
Week 9:  Jordan normal form and its applications
Week 10: Sesqui-linear forms
Week 11: Inner Product Spaces
Week 12: Spectral Theorem

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