A general outline of the genesis of vector space theory
type: Item
instance of: scholarly article
title: A general outline of the genesis of vector space theory
zbMATH Open document ID: 0834.01006
author: Jean-Luc Dorier
published in: Historia Mathematica
publication date: 1996-04-01T00:00:00Z
full work available at URL: unige:16642
review text: The paper is what the title says, an outline; it is not an everywhere thorough investigation. Though vector space theory is quite common today, its history is not yet well known. This is the reason, why the list of references is mainly based on sources, there really does not exist much secondary literature. From antiquity up to the end of the 19th century linear methods were adopted in different branches of mathematics but there was no unification, no focusing on the main points, there was no structure like definition. One of these branches was the theory of linear equations. Though in 1750 Euler and Cramer made first steps in direction of a theory of determinants, they neglected undetermined and inconsistent systems of linear equations. Only between 1840-1879 the ``rank'' of a determinant was defined and became a central point in the description of linear equations, the final results are due to Frobenius. A second branch was the concept of vector in geometry. In this context there are mentioned Möbius and Bellavitis, Bellavitis being the first to define an addition of vectors in space. But, according to the author, the elaboration of geometric calculus seems to have had more influence on the development of vector analysis than on the theory of vector space (p. 237). Grassmann's Ausdehnungslehre is only regarded as an isolated singularity (p. 241-246). Further impulses are due to ``linear algebra in the dialectic exchanges between geometry and algebra'' (p. 237-241), these are mainly the contributions to the theory of transformations, i.e. the results of Euler, Lagrange, Gauss, Cayley and Sylvester. In the second half of the 19th century there existed a geometry of \(n\) dimensions, developed on the basis of analytic geometry and the theory of determinants and matrices. The first axiomatic approaches were made by Peano and his school and Weyl (1888, 1918). In 1893 Dedekind gave a precise definition of a linear structure and in 1910 Steinitz contributed the definitio…
reviewed by: Karin Reich
Mathematics Subject Classification ID: 01A50
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 01A60
zbMATH DE Number: 801052
zbMATH Keywords: linearity
zbMATH Keywords: linear combination
zbMATH Keywords: rank
MaRDI profile type: Publication
OpenAlex ID: W2025234661
Wikidata QID: Q55890534
cites work: Q5824372
cites work: Algèbre commutative
cites work: Q5549766
cites work: Q3935670
cites work: Q4282465
cites work: Q4335840
cites work: Hermann Grassmann and the Creation of Linear Algebra
cites work: Hermann Grassmann and the Prehistory of Universal Algebra
cites work: Q4323305
cites work: Q5556367
cites work: Q5349018
cites work: Q3056945
cites work: Q5536198
cites work: Q3346268
cites work: H. Grassmann's 1844Ausdehnungslehreand Schleiermacher'sDialektik
cites work: Q4773657
cites work: Benjamin Peirce's Linear Associative Algebra
cites work: George Peacock and the British origins of symbolical algebra
cites work: The art and the science of British algebra: a study in the perception of mathematical truth
cites work: On the exterior calculus of invariant theory
cites work: Q3257074
DOI: 10.1006/HMAT.1995.1024
description: scientific article; zbMATH DE number 801052