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 SO(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.