MATE-1107 Linear Algebra II

Review of the previous course (Mate-1105) in more depth: Vector spaces, subspaces, linear combinations, bases and dimension, Linear transformations, core and image, matrix representation of a linear transformation, coordinate change matrix, dual space, elemental matrices and linear equation systems, determinants, their characterization as a multilinear form, values and vectors, diagonalizability, invariant subspaces, the Cayley-Hamilton Theorem, Spaces with an Internal Product: Adjoint operator, normal, self-adjoint, unitary and orthogonal operators, orthogonal projection and spectral theorem, bilinear and quadratic forms. Applications on the theory of relativity: Einstein’s principle of relativity, Lorentz’ transformations. Jordan Canonical Form: Jordan normal form, minimal polynomial. Multilinear algebra and tensors: Tensors on a vector space, examples and applications.