Items list
Description
Group action on a set, orbits, stabilizers. Burnside Enumeration Theorem and applications. Matrix groups, isometries of Euclidean spaces, symmetry groups, finite symmetry groups in the plane and in space, Platonic solids. Modules, linear mappings, kernel, image, isomorphism theorems, direct products and sums, dual module, free modules. Modules over regions of principal ideals. Invariant factors, rational normal matrix form, elementary divisors, Jordan matrix form, matrix norms, matrix exponentials. Chained complexes modules, homology, cohomology, associative morphism. Multilinear forms and duality in vector spaces, tensor product of vector spaces, tensor product of linear maps, Kronecker product of matrices, symmetric tensors, antisymmetric tensors. Tensor algebra, universal property. Exterior algebra, dimension, functionalism. Adjoint mapping, special types of matrices (symmetric, Hermitian, orthogonal, monadic, normal), eigenvalues and diagonalization. Hilbert basis theorem, Gröbner bases, Buchberger algorithm.
