Linear Algebra and Geometry

Abstract

Part 1 Linear spaces and linear mappings: linear spaces basis and dimension linear mappings matrices subspaces and direct sums quotient spaces duality the structure of a linear mapping the Jordan normal form normed linear spaces functions of linear operators complexification and decomplexification the language of categories the categorical properties of linear spaces. Part 2 Geometry of spaces with an inner product: on geometry inner products classification theorems the orthogonalization algorithm and orthogonal polynomials Euclidian spaces unitary spaces orthogonal and unitary operators self-adjoint operators self-adjoint operators in quantum mechanics the geometry of quadratic forms and the Eigenvalues of self-adjoint operators three-dimensional Euclidean space Minkowski space symplectic space Witt's theorem and Witt's group Clifford algebras. Part 3 Affine and projective geometry: affine spaces, affine mappings and affine coordinates affine groups affine subspaces convex polyhedra and linear programming affine quadratic functions and quadrics projective duality and projective quadrics projective groups and projections Desargues' and Pappus' configurations and classical projective geometry the Kahler metric algebraic varieties and Hilbert polynomials. Part 4 Multilinear algebra: tensor products of linear spaces canonical isomorphisms and linear mappings of tensor products the tensor algebra of a linear space classical notation symmetric tensors skew-symmetric tensors and the exterior algebra of a linear space exterior forms tensor fields tensor products in quantum mechanics.