GEOMETRY B

Linear maps and matrices in the euclidean plane: projections, reflections, rotations, isometries. Group theory. Real and complex vector spaces.
Linear maps: kernel and image, the dimension's formula. Basis change and matrices. Matrices of linear maps on finite dimensional vector spaces.
Invariant subspaces, eigenvalues, eigenvectors. Diagonalization of operators. Orthogonal matrices and operators. Systems of ordinary differential equations with constant coefficients.
Bilinear forms and scalar products. Spectral theorem. Diagonalization of symmetric matrices using orthogonal matrices. Classification of quadric surfaces in the space.