An Introduction to Finite Dimensional Vector Spaces

This chapter introduces vector spaces of finite dimension. It attempts to present a self-contained and focused introduction to the essential concepts and methods for vector spaces of finite dimension. The applications are numerous, for example numerical linear algebra, finite Markov chains, multivariate optimisation, machine and statistical learning, graph theory and finite difference methods, to name just a few. The chapter also presents a precise and compact introduction to finite-dimensional vector spaces and linear transformations between vector spaces. Mappings between vector spaces are at least as interesting as vector spaces themselves. An important property of linear transformations is that they map linearly dependent subsets into linearly dependent subsets. An interesting remark is that the set of all linear transformations between two given vector spaces is itself a vector space.