Dorier, Jean-Luc Basis and dimension – from Grassmann to van der Waerden. (English) Zbl 0907.01008 Schubring, Gert (ed.), Hermann Günther Graßmann (1809-1877): visionary mathematician, scientist and neohumanist scholar. Papers from the sesquicentennial conference on 150 years of “Lineale Ausdehnungslehre” - Hermann G. Graßmann’s work and impact, Lieschow / Rügen, Germany, May 23–28, 1994. Dordrecht: Kluwer Academic Publishers. Boston Stud. Philos. Sci. 187, 175-196 (1996). This article examines the concepts: basis, dimension, generation or generators, independence and independence, unicity of coordinates, subspace, rank. It covers the two initial contribution of Grassmann in 1844 and 1862, contrasting the inherent independence of the generators, and the invariance of dimension from the exchange process, and the introduction of coordinates by the ratios of the lengths of collinear vectors with the later treatment where the isomorphism with \({\mathbb R}^n\) is the foundation for the theory. Peano’s axiomatics for \({\mathbb R}^n\) of 1888: here a maximum number \(n\) of independent vectors is used for the basis, uniqueness of coordinates is proved, there is no hint of “generators” nor is the query of a spanning set of less than \(n\) elements raised. His work flowed to Burali-Forti and Marcolongo in 1909, but not, so Dorier asserts, to Hermann Weyl’s work of 1918 and Steinitz in 1910. Both these have axioms, with an approach to dimension similar to that of the Italians. The theory of linear equations from Euler and Cramer in 1750 led to the theory of elimination and the representation of solutions through determinants, even for the case where the number \(r\) of non-eliminated equations is less than the number \(n\) of equations given. It was Frobenius in 1875 who introduced rank \(r\) as the largest order of non-zero minors, who showed the equivalence of the dependence of \(n\)-tuples (vectors) with that of the coefficients of the linear equations. In 1886 Capelli and Garbieri identified the rank with the number of non-zero diagonal coefficients in the triangular form of the equations. Modern algebra applies these concepts of basis, dependence and dimension in the field theory of Dedekind, who introduced our modern methods in his 11-th Supplement to Dirichlet’s lectures on number theory. And it is in (algebraic) number theory that Steinitz stated the “Italian” definition of dependence, that \(n\) are independent but \(n+1\) are dependent. Finally there came the unified general approach of van der Wareden in 1930-1931 in his Moderne Algebra. The article concentrates on the few concepts; it is in the main clear and concise, well based; there are several places where retention of older terminology obscures rather than reveals.For the entire collection see [Zbl 0853.00027]. Reviewer: J.J.Cross (Melbourne) Cited in 1 ReviewCited in 4 Documents MSC: 01A55 History of mathematics in the 19th century 15-03 History of linear algebra Keywords:Grassmann; multilinear algebra; basis; dimension Biographic References: Grassmann, H. G.; Van der Waerden, B. L. PDFBibTeX XMLCite \textit{J.-L. Dorier}, Boston Stud. Philos. Sci. 187, 175--196 (1996; Zbl 0907.01008)