MA5310    Linear Algebra

Course contents:

 

Systems of Linear Equations, Matrices and Elementary Row Operations, Row-Reduced Echelon Matrices .

Vector Spaces, Subspaces, Bases and Dimension, Ordered basis and coordinates.

Linear transformations, Rank-Nullity Theorem, The algebra of linear transformations, Isomorphism, Matrix representation of linear transformations, Linear Functionals, Annihilator, Double dual, Transpose of a linear transformation .

Characteristic Values and Characteristic Vectors of linear transformations, Diagonalizability, Minimal polynomial of a linear transformation, Cayley-Hamilton Theorem, Invariant Subspaces, Direct-sum decompositions, Invariant Direct sums, The primary decomposition theorem, Cyclic subspaces and annihilators, Cyclic decomposition, rational, Jordan forms.

Inner Product Spaces, Orthonormal Basis, Gram-Schmidt Theorem.

 

Text Books:

1.      K. Hoffman and R. Kunze: Linear Algebra, 2nd Edition, Prentice Hall of India, 2005.

2.      M. Artin: Algebra, Prentice Hall of India, 2005.

References:

1.    I. N. Herstein: Topics in Algebra, 2nd Edition, John-Wiley, 1999.

2.    S. Axler: Linear Algebra Done Right, 2nd Edition, Springer UTM, 1997.

3.    S. Lang: Linear Algebra, Springer Undergraduate Texts in Mathematics, 1989.

4.    S. Kumaresan: Linear Algebra: A Geometric Approach, Prentice-Hall of India, 2004.