Linear Algebra

Semester I (PS01CMTH24)
old course

Syllabus

Unit-I Quick review of vector spaces, examples of sequence and function spaces. Linear spans; linear dependence/independence and basis, Examples of finite dimensional and infinite dimensional vector spaces, quotient space and its dimension. Dual space, dual basis, dimension of the annihilator. Solution of the system of simultaneous linear homogeneous equations.
Unit-II Definitions and examples of algebra, algebra analog of Cayley theorem. Minimal polynomial of a linear transformation. Rank of a linear transformation. Characteristic roots, characteristic vectors and results related to characteristic vectors. Matrix associated with a linear transformation on finite dimensional vector space. Isomorphism between the space of linear transformations and the space of matrices. Similarity of matrices and similarity of linear transformations.
Unit-III Relation of the minimal polynomials of a linear transformation and its induced linear transformation on a quotient space, triangular matrix associated to a linear transformation. Nilpotent linear transformation. Canonical matrix associated to a nilpotent linear transformation. Existence and uniqueness of invariants of a nilpotent linear transformation. Jordan form of a linear transformation.
Unit-IV                             Trace and its applications, Jocobson’s lemma. Transpose of a matrix. Definition of the determinant of a matrix, determinant of a triangular matrix, a matrix with equal rows, a product of matrices. Application of determinant: regularity of a matrix, Cramer’s Rule to solve system of simultaneous non-homogeneous linear equations. Quadratic forms: diagonalization of a symmetric matrix. Symmetric matrix associated to a quadratic form, classification of quadratics.


Reference Books

  1. Herstein I. N., Topics in algebra, Wiley Eastern Ltd., New Delhi, (2nd Edition, 1975).
  2. Kwak J. H., Hong S., Linear Algebra, (Second edition), Birkhauser.
  3. Kumaresan, S., Linear Algebra: A Geometric Approach, Prentice Hall of India, 2000.
  4. Simmons G. F., Introduction to Topology and Modern Analysis, McGraw-Hill Co., Tokyo, 1963.
  5. Helson, H., Linear Algebra, (Second Edition), Hindustan Book Agency, TRIM-4, 1994.
  6. Ramachandra Rao A. and Bhimasankaram P., Linear Algebra (Second Edition), Hindustan Book Agency, TRIM.

Lecture Notes

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