SYLLABUS

ALGEBRA AND ANALYTICAL GEOMETRY II

Four credits

COURSE OBJECTIVES

The purpose is to provide students with more knowledge in higher linear algebra and analytical geometry such as: eigenvalues and eigenvectors of matrices and linear operators. Affine spaces.Euclidean spaces. Bilinear forms; complex and real quadratic forms. Classification of second-order curves and surfaces.

PREREQUISITES

- Algebra and Analytical geometry I

DETAILED COURSE OUTLINE

1. Canonical forms of linear operators and square matrices

1.1. Eigenvalues, Eigenvectors and Eigenspaces of linear operators and square matrices

1.2. Characteristic polynomials. Hamilton-Calley’s theorem.

1.3. Diagonalisation

2. Affine spaces

2.1. Affine spaces and affine coordinates

2.2. Planes in affine spaces

2.3. Affine mappings. Isomorphisms of affine spaces

2.4. Affine transformations. Matrices of affine transformations.

3. Euclidean spaces.

3.1. Inner products and euclidean spaces

3.2. Orthogonality. The Gram–Schmidt’s orthogonalization process.

3.3. Orthogonal and orthonormal bases. Least squares method. Distance from a vector to a finite dimensional subspace.

3.4. Linear operators over euclidean spaces. Isomorphisms of euclidean spaces. Adjoint operators.

3.5. Orthogonal operators and orthogonal matrices.

3.6. Symmetric and antisymmetric matrices. Orthogonal diagonalization of real symmetric matrices.

4. Bilinear and quadratic forms.

4.1. Bilinear forms. Matrices and ranks of bilinear forms. Symmetric bilinear forms. Change of bases.

4.2. Quadratic forms. Matrices and ranks of quadratic forms. Change of bases.

4.3. Canonical forms of quadratic forms: Definition. Reduction of quadratic forms to canonical forms by Lagrange algorithm.

4.4. Real quadratic forms. Law of inertia. Definite quadratic forms (positive or negative) and Sylvester’s criterion. Reduction of real quadratic forms to canonical forms by orthogonal operators.

4.5. Second-degree curves. Reduction of equations of second-degree curves to canonical forms. Classification of second-degree curves.

COURSE MATERIALS

1. Ngoâ Thuùc Lanh. Linear algebra, HaNoi, 1970 (in Vietnamese).

2. Serge Lang. Algebra (Part III), HaNoi, 1978 (in Vietnamese).

3. Vaên Nhö Cöông, Kieàu Huy Luaân, Higher Geometry, NXB GD, 1978 (in Vietnamese).

4. David C.Lay. Linear algebra and its applications, Addision–Wesley Publising Co.,1994.

5. Kenneth Hoffman & Ray Kunze. Linear algebra, Prentice Hall, Inc.,1971.

6. V.A.Ilyin & E.G.Poznyak. Linear algebra, Moscow, Mir Publishers, 1976.

7. Roger Godement. Algebra, Hermann, Paris, 1968.