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.