LINEAR ALGEBRA

Vector spaces: Basis and dimension, Direct sums.

Determinants: Theory of determinants, Cramer’s rule.

Linear transformations: Rank-nullity theorem, Algebra of linear transformations, Dual spaces. Linear operators, Eigenvalues and eigenvectors, Characteristic polynomial, Cayley- Hamilton theorem, Minimal polynomial, Algebraic and geometric multiplicities, Diagonalization, Jordan canonical Form.

Symmetry: Group of motions of the plane, Discrete groups of motion, Finite groups of S0(3).

Bilinear forms: Symmetric, skew symmetric and Hermitian forms, Sylvester’s law of inertia, Spectral theorem for the Hermitian and normal operators on finite dimensional vector spaces.

Linear groups: Classical linear groups, SU₂ and SL ₂(R).

Books

• Artin, M., Algebra, Prentice-Hall of India, 1994.
• Herstein, I. N., Topics in Algebra, Vikas Publications, 1972.
• Strang, G., Linear Algebra and its Applications, Third Edition, Saunders, 1988.
• Halmos, P., Finite dimensional vector spaces, Springer-Verlag (UTM), 1987.