### MA5310 Linear Algebra

#### Course Details

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.

#### Course References:

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.