Geometry 2

1. Endomorphisms of vector spaces and their canonical forms
Eigenvalues ​​and eigenvectors of an endomorphism. Diagonalizable endomorphisms. Minimum polynomial. Cayley Hamilton theorem. Jordan's canonical form ..
2. Euclidean vector spaces
Internal products in real and complex vector spaces. Orthonormal bases; Gram-Schmidt procedure. Isometries and orthogonal group. Symmetric endomorphisms. Real spectral theorem. The complex case.
3. Bilinear and quadratic forms
Multi-linear forms. Bilinear forms; congruent matrices. Canonical reduction of a quadratic form. Real quadratic forms. Sylvester's theorem. Complex quadratic forms.
4. Geometry in n-dimensional spaces on an arbitrary field
Euclidean affine spaces. Orthonormal coordinate systems. Linear subspaces and their representations. Distances, angles. Coordinate changes and transformations. Projective spaces. Projective linear subspaces and their representations. Grassmann's formula. Fundamental theorem of projective geometry. The affine complementary space of a hyperplane. Projectivities and affinities.
5. Quadrics and conics
Hyperquadrics from the real / complex projective point of view: singular points; reducibility, classification. Hyperquadrics in the affine space. Hyperquadrics in Euclidean space.