Page 4 - DMTH502_LINEAR_ALGEBRA

SYLLABUS

Linear Algebra

Objectives: This course is designed for theoretical study of vector spaces, bases and dimension, subspaces, linear transformations,
dual spaces, Elementary Canonical forms, rational and Jordan forms, inner product spaces, spectral theory and bilinear forms.
It should be noted that the successful student will be able to prove simple theorems in the subject.

Sr. No. Description

1 Vector Space over fields, Subspaces, Bases and Dimension, Coordinates, Summary

of Row-Equivalence, Computation Concerning Subspaces

2 Linear Transformations, The algebra of linear transformations, The transpose of a
linear transformation, Isomorphism, Representation of Transformation by matrices

3 Linear Functional, The double dual, Introduction and Characteristic Values,

Annihilating Polynomials

4 Invariant Subspaces, Simultaneous triangulation, Simultaneous diagonalization,

Direct-Sum Decompositions

5 Invariant Direct Sums, The Primary Decomposition Theorem, Cyclic Subspaces and
Annihilators, Cyclic Decomposition and the rational Form

6 The Jordan Form, Computation of Invariant Factors, Semi-Simple Operators

7 Inner product, Inner Product Space, Linear Functional and Adjoints, Unitary

Operators, Normal Operators

8 Introduction, Forms on Inner Product Spaces, Positive Forms, More on Forms

9 Spectral Theory, Properties of Normal operators

10 Bilinear Forms, Symmetric Bilinear Forms, Skew-Symmetric Bilinear Forms,
Groups Preserving Bilinear Forms