Introduction to Vector Spaces, Vector Algebras, and Vector Geometries

TL;DR

This paper provides an introductory overview of vector spaces, algebras, and geometries, emphasizing quotient spaces, tensor products, and applications like exterior algebra and projective geometry.

Contribution

It offers a comprehensive introduction to advanced concepts in linear algebra and geometry, including quotient spaces, exterior and symmetric algebras, and projective geometries.

Findings

Introduction of quotient spaces and their use in algebraic constructions

Explanation of exterior and symmetric algebras of vector spaces

Overview of affine, projective, and scalar product geometries

Abstract

An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space. Affine geometries are introduced and generalized by projective completion. General projective geometries are briefly introduced. Tensor products and multilinear functions are treated. The exterior algebra of a vector space and that of its dual are used in treating linear geometry and Grassmann's regressive product is treated. Scalar product spaces, orthogonality, and the Hodge star based on a general basis are covered.