Algebra IV Euclidean spaces

This course is an introduction to bilinear algebra, covering Euclidean and Hermitian spaces. It covers isometries, duality, quadratic forms and endomorphisms.

Euclidean spaces:

scalar product, Cauchy-Schwarz, Euclidean norm and distance, triangular inequality, parallelogram equality, Pythagorean theorem. Orthonormal basis.

Gram-Schmidt orthonormalization algorithm. Vector angles, line angles, center angle theorem and cocyclicity. Orthogonal subspaces.

Determinant in an orthonormal basis and volume. Orientation.

Orthogonal projections (application to the method of least squares).

Linear isometries, orthogonal matrices, orthogonal group and orthogonal special. Examples of isometries: rotations, symmetries. Classification of isometries in dimensions 2 and 3.

Isometries preserving a regular polygon of the plane

Duality.

Definition of dual and bidual. Orthogonal of a subspace (in the sense of duality), dual basis, antehedral basis. Correspondence between hyperplanes and linear forms, duality between parametric and Cartesian description of a subspace. Adjoint of an endomorphism. Matrix writing, link with transpose.

Symmetrical bilinear forms on an R-e.v.

Matrix of a bilinear form. Bilinear form as linear application between space and its dual. Core and rank of a bilinear form. Isotropic vectors. Quadratic form. Existence of orthogonal bases. Gauss reduction algorithm. Sylvester's inertia theory, signature of a quadratic form. Classification of real quadratic forms.

Interpretation of duality in a Euclidean space. Symmetric and orthogonal endomorphisms in a Euclidean space. Link with the adjoint. Associated quadratic form. Diagonalization of symmetric matrices in an orthonormal basis. Simultaneous diagonalization of two symmetric forms, one of which is positive definite.

Hermitian sesquilinear forms and Hermitian spaces.

Review of concepts seen in the real case: definition, matrix, Hermitian quadratic form, signature and Sylvester's inertia theorem in this framework. Hermitian spaces, definitions, similarities and differences with Euclidean spaces, unitary group, self-adjoint endomorphisms. Notion of complexification and real forms.

Normal endomorphisms:

 reduction, with applications to symmetric, antisymmetric, orthogonal, unitary and self-adjoint matrices.

L1 linear algebra (HAX102X and HAX202X)

and HAX301X: Algebra III Reduction of endomorphisms

Recommended prerequisites: L1 maths

Hourly volumes :

            CM: 30

            TD : 30

            TP :

            Terrain :