Advanced Vector Spaces (AdVec)

Knowledge: Central definitions and theorems from the subjects mentioned in the description of contents. In particular, the following notions are considered central:

Linear dependence, basis, dimension, quotient space, quotient map, invariant subspace, rank, nullity, dual space, dual basis, adjoint map, direct sum, projection, idempotent map, bilinear form, alternating form, quadratic form, positive definite form, non-degenerate, tensor product, multilinear form, wedge product, determinant, trace, eigenvalue, eigenvector, eigenspace, spectrum, spectral radius, geometric multiplicity, algebraic multiplicity, characteristic polynomial, diagonability, flag, inner product, Hilbert space, self-adjoint map, normal map, unitary map, nilpotent map, cyclic vector, generalized eigenspace, operator norm, spectral radius, positive definite map, principal minors, leading principal minors.

Skills/Competencies:

To follow and reproduce proofs of statements within the subjects mentioned in the description of contents and involving the notions mentioned above.

To understand the relationships between the different subjects of the course

To prepare and give a coherent oral presentation of a random mathematical topic within the curriculum of the course.