Algebra IV: Euclidean Spaces

This course is an introduction to bilinear algebra and will cover Euclidean and Hermitian spaces. It will address topics such as isometries, duality, quadratic forms, and endomorphisms.

Euclidean spaces:

Scalar product, Cauchy-Schwarz inequality, Euclidean norm and distance, triangle inequality, parallelogram inequality, Pythagorean theorem. Orthonormal basis.

Gram-Schmidt orthogonalization algorithm. Angles between vectors, angles between lines, the central 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, the orthogonal group, and the special orthogonal group. Examples of isometries: rotations, symmetries. Classification of isometries in 2 and 3 dimensions.

Isometries that preserve a regular polygon in the plane

Duality.

Definition of the dual and the bidual. Orthogonal to a subspace (in the sense of duality), dual basis, antidual basis. Correspondence between hyperplanes and linear forms; duality between parametric and Cartesian descriptions of a subspace. Adjoint of an endomorphism. Matrix representation; connection to the transpose.

Symmetric bilinear forms on an R-vector space

Matrix of a bilinear form. Bilinear form as a linear map between a 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. Connection to 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 the concepts covered in the real-world example: definition, matrix, Hermitian quadratic form, signature, and Sylvester’s inertia theorem in this context. Hermitian spaces, definitions, similarities and differences with Euclidean spaces, the unitary group, and self-adjoint endomorphisms. The concepts 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: First-year math

Hourly volumes:

            CM: 30

            TD: 30

            Practical Work:

            Lot: