Vector Spaces, Affine Spaces, and Metric Spaces

Abstract

The chapter gives a short overview of the most important concepts in linear algebra, affine spaces, and metric spaces. It is not intended as a course, but as a point of reference and a brush up.

First, we present the basic concepts of linear algebra: vector space, subspace, basis, dimension, linear map, matrix, determinant, eigenvalue, eigenvector, inner product. This should all be familiar concepts, but what might be less familiar is the abstract view where the basic concepts are vector spaces and linear maps while coordinates and matrices become derived concepts. The last subject is the singular value decomposition which is used for mesh simplification and in the ICP algorithm for registration.

The next area is affine spaces where we only give the basic definitions: affine space, affine combination, convex combination, and convex hull.

Finally we introduce metric spaces which makes the concepts of open sets, neighborhoods, and continuity precise.