Geometry 2

1. Vector space endomorphisms and canonical forms
Eigenspaces and invariant subspaces of an endomorphism (of a square matrix). Characterization of diagonalizable and triangularizable endomorphisms. Jordan's canonical form. Polynomials and eigenvalues. Cayley-Hamilton's theorem. Minimal polynomial.
2. Euclidean vector spaces
Inner products in real and complex vector spaces. Norm; angles; orthogonality; orthonormal bases; Gram-Schmidt process. Isometries and the orthogonal group. Symmetric endomorphisms and their properties (the real spectral theorem). A look at the complex case.
3. Bilinear and quadratic forms
Multilinear forms. Bilinear forms; congruent matrices. The quadratic form associated to a symmetric bilinear form. Conjugate bases; reducing a quadratic form to canonical form. Real quadratic forms. Sylvester's theorem. Definite, semi-definite, indefinite real quadratic forms. Pseudo-euclidean vector spaces. Complex quadratic forms.
4. Geometry in n-dimensional spaces over an arbitrary field
Euclidean affine spaces. Orthonormal frames. Linear varieties and their analytic representations. Distances, angles, areas, volumes, hypervolumes. Changes of coordinates and transformations. Euler's theorem and canonical form of rotations.
Projective spaces. Motivations. Homogeneous projective coordinates. Cross ratio. Projective linear varieties and their representations. Grassmann's formula. The main theorem of projective geometry. The affine space complementing a hyperplane. Projective transformations and affine transformations. Idea of geometry according to F. Klein.
5. Quadrics and conics
Conics in the Euclidean, affine, and projective planes. Quadric hypersurfaces from the real/complex projective point of view: singular points; reducibility; linear spaces; polarity; the nature of points of a real quadric surface; classifications. Quadric hypersurfaces in affine space. Projective closure. Behaviour with respect to the hyperplane at infinity. Affine classification from the real and complex point of view. Quadric hypersurfaces in the Euclidean space. Orthogonal invariants. Classification of conics and quadric surfaces from the euclidean-metric point of view.
6. Duality
Dual of a vector space. Dual basis. Transpose of a linear map. Annihilators and their properties. The duality principle in projective geometry illustrated trough elementary examples. Envelopes.