Geometry and Linear Algebra WM-MA-GAL

1. Vector Euclidean spaces: dot product, vector Euclidean space, Cartesian Euclidean space, Schwartz inequality, norm, orthogonality, angle between vectors, the law of cosines, orthogonal basis, Gram-Schmidt orthogonalization, theorem on orthogonal projection,, Gram matrix and determinant, geometric interpretation of the Gram determinant: parallelepiped volume, orientation of a vector space, vector product.
2. Endomorphisms of Euclidean spaces: self-adjoint endomorphisms,
theorem on diagonalization of self-adjoint endomorphisms, orthogonal matrices, linear isometries, classification of linear isometries in Euclidean spaces.
3. Affine spaces, free vectors, the dimension of affine space, affine subspaces: line, hyperplanes, affine combinations and affine spans, independent affine systems, base systems, point basis, barycentric coordinates, affine transformations, affine isomorphisms.
4. Euclidean spaces: affine Euclidean spaces, orthogonal projection of Euclidean space on subspace, orthogonal symmetry, isometries of Euclidean spaces.
5. Quadratic forms: symmetric bilinear functionals, quadratic form and its polar form, congruent matrices, rank of the quadratic form, Lagrange's diagonalization theorem, Jacobi's condition, positively and negatively defined quadratic form, Sylvester's inertia theorem.
6. Quadric hypersurface and its classification.