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).