Algebra IV Euclidean spaces

This course is an introduction to bilinear algebra and will cover Euclidean and Hermitian spaces. It will cover everything related to isometries, duality, quadratic forms, and endomorphisms.

Euclidean spaces:

scalar product, Cauchy-Schwarz, norm and Euclidean distance, triangle 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 least squares method).

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

Isometries preserving a regular polygon in the plane

Duality.

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

Symmetric bilinear forms on an R-e.v.

Matrix of a bilinear form. Bilinear form as linear applications between the space and its dual. Kernel 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 Euclidean space. Symmetric and orthogonal endomorphisms in 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 context. Hermitian spaces, definitions, similarities and differences with Euclidean spaces, unitary group, self-adjoint endomorphisms. Concept of complexification and real forms.

Normal endomorphisms:

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

Linear Algebra I (HAX102X and HAX202X)

and HAX301X: Algebra III Reduction of Endomorphisms

Recommended prerequisites: L1 math

Hourly volumes:

            CM: 30

            TD: 30

            TP:

            Land: