Linear algebra and geometry II* 1000-112bGA2*

1. Endomorphisms of linear spaces. Matrix of an endomorphism in a given basis. Determinant and trace of an endomorphism. Eigenvectors eigenspaces and eigenvalues. Characteristic polynomial. Diagonal matrices, diagonalizable matrices and endomorphisms, criteria of diagonalizability. Jordan normal form.

2. Affine spaces as cosets in a linear spaces. Affine combinations, affine independency, affine bases, barycentric coordinates. Affine coordinate systems. Affine maps and corresponding linear mappings. Matrices of affine maps. Isomorphisms of affine spaces, every affine space is isomorphic to K^n. Axomatic definition of affine spaces.

3. Linear functionals, dual spaces. Dual bases, coordinates of a functional in a dual basis. Isomorphism of a finite-dimensional vector space and its dual. Dual maps an their matrices in dual bases.

4. Scalar product. Schwarz inequality. Inner product spaces. Orthogonal complement of a subspace. Othogomal projections and symmetries. Orthogonal and othonormal bases, coordinates in such bases. Gram-Schmidt process. Sylvester's criterion. Gram matrix and its properties.

5. Euclidean spaces (affine spaces with a scalar product). Distance between points, distance between a point and a subspace. Measure, volume of parallelotopes and simplices. Angles. Orientation. Cross product.

6. Maps of euclidean spaces preserving scalar product, isomorphisms of euclidean spaces. Orthogonal matrices. Isometries. Self-adjoint maps. Digonalization of symmetric real matrices via orthogonal matrices.

7. Hermitian product. Isomorphisms of spaces with hermitian product, unitary matrices.

8. Bilinear forms, symmetric forms. Matrix of a bilinear form in a basis, congruence of matrices. Non-degenerate forms. Orthogonal complement of a subspace of a space with non-degenerate form. Orthogonal bases. Every finite dimensional space with a symmetric bilinear form has an orthogonal basis (for char K<>2). Sylvester's law of inertia. Congruence classes of real and complex matrices. Quadratic forms and methods of diagonalization of quadratic forms.

9. Polynomials and polynomial functions. Polynomial functions on affine spaces. Algebraic sets, hypersurfaces. Classification of real and complex degree 2 hypersurfaces. Detailed description in cases R^2 and R^3. Classification of degree 2 hypersurfaces in R^n up to isometry.