**LINEAR
ALGEBRA**

** Vector spaces:**
Definition, Basis and dimension, Direct sums.

**Linear transformations:**
Definition, Rank-nullity theorem, Algebra of linear transformations, Dual spaces,
Matrices.

** Systems of linear equations: **elementary theory of determinants, Cramer's rule. Eigenvalues and eigenvectors, the characteristic polynomial, the Cayley-Hamilton Theorem, the minimal polynomial, algebraic and geometric multiplicities, Diagonalization, The 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.

**Recommended Books**

- Artin, M., Algebra, Prentice-Hall of India, 1994.
- Halmos, P., Finite dimensional vector spaces, Springer-Verlag (UTM), 1987.
- Hoffman, K. and Kunze, R., Linear Algebra, 2nd edition, Prentice-Hall of India, 1992