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What is the origin of calling a vector space over a field $F$ endowed with multiplication an algebra? Tried searching, but not surprisingly Google likes to drop the article and just bring me to the etymology of algebra sans article. Thanks.

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Actually, it happened in the reverse order, algebras came first, and vector spaces only later. For the vector space story see When did people start viewing a matrix as a linear transformation between two vector spaces? Peano gave the modern axiomatization of them only in 1888, and he called them linear systems. But the use of "an algebra" in essentially modern sense goes back to 1870, when Benjamin Peirce (the father of C.S. Peirce known for inventing the material conditional and giving Boolean algebra its modern form) introduced it in his memoir Linear Associative Algebra read to the US National Academy (published in 1881). He explained it thus:

"All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole. Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra. In all other algebras both relations must be combined, and the algebra must conform to the character of the relations. The symbols of an algebra, with the laws of combination, constitute its language; the methods of using the symbols in the drawing of inferences is its art; and their interpretation is its scientific application. This three-fold analysis of algebra is adopted from President Hill, of Harvard University, and is made the basis of a division into books" (boldface mine).

As the quote indicates the use of "an algebra" more broadly for a system of objects with "addition" and "multiplication" predates Peirce. "Linear associative algebra" specifically is what we today call an algebra over $\mathbb{C}$ in the narrow sense. An alternative term used at the time was "hypercomplex number system" and the field was a vibrant area of research since Hamilton's discovery of quaternions and Grassman's "geometric algebra". Cayley, Sylvester, Frobenius, Veblen, Wedderburn, etc., were working in it in the second half of 19th early 20th century. The famous classification of division algebras over $\mathbb{R}$ (Frobenius theorem) belongs to this period.

Various connections to geometry, already present in Hamilton and Grassman, were also explored at length, especially after Klein's promotion of Erlangen Program, see What was the motivation for Minkowski spacetime before special relativity? In addition to contemporary algebraic geometry the field was an important incubator of ideas and techniques that were later shaped into modern abstract algebra by Emmy Noether and van der Waerden in 1920s. That is when the definition was extended beyond $\mathbb{C}$ to algebras over arbitrary fields. For more on this later story see What was the evolution of "basis" and "generating set" in algebra?

An authoritative historical work on the early days of linear associative algebras is Parshall's Joseph H. M. Wedderburn and the Structure Theory of Algebras. A very insightful and nicely written (albeit less historically rigorous) account of 19th century work on hypercomplex numbers and Kleinian geometries is Yaglom's Felix Klein and Sophus Lie. Its scope is much broader than its title, it explores the interplay of algebraic and geometric ideas also in works of Galois, Poncelet, Hamilton, Grassmann, Cayley, Peirce, Clifford, etc., and gives biographical details and extensive references to original and secondary sources.

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Between algebras and vector spaces there was (and still is, as taught in some physics courses or in those known as Vector Calculus at many American universities) the algebra of vectors developed by Oliver Heaviside to serve the needs of physics. It appeared in his paper "On the forces, stresses, and fluxes of energy in the electromagnetic field", Philosophical Transactions of the Royal Society (1892), pp. 521-574 and treats rigorously the now-familiar operations on vectors such as scalar or vector product, also divergence and curl.

The commentary by Ido Yavetz in Chapter 49 (on Heaviside's electrical papers) in "Landmark writings in Western Mathematics 1640-1940", edited by Ivor Grattan-Guinness, Elsevier, 2005, explicates the algebraic origins of Heaviside's work:

"The remarkable thing about vector algebra as we currently know it, is that it did not emerge from a direct attempt to formalize such widely familiar applications. Instead, it made its first appearance in the context of Hamilton's highly innovative and idiosyncratic invention of quaternions. (...) In their quaternionic context, however, vectors possessed formal properties that proved awkward for application to physical problems. it was then left for Gibbs and Heaviside to extract the vector from its quaternionic foundations and establish it on its own formal grounds."

Added on November 11, 2016: A good reference is: Michael J. Crowe, A history of vector analysis: the evolution of the idea of a vectorial system. Originally published by Notre Dame University Press, 1962; Dover reprints 1985 and 1994.

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  • $\begingroup$ So is that to say Hamilton's expansion of $\mathbb{C}$ was understood as distinct from vectors, and it was Heaviside who reinterpreted it in the context of vector spaces by introducing extra operations to the vector space? $\endgroup$ – AJY Aug 19 '16 at 17:31
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    $\begingroup$ I am not an expert on this, but as far as I know, Hamilton was interested in finding a group which would generalize complex numbers. He considered the relations $i^2=j^2=k^2=ijk=-1$ the essence of his discovery, which is what he carved into the side of the Broom Bridge in Dublin in 1843. So he focused on algebraic properties of quaternions. I would say Heaviside introduced the first example of a proto-vector space, since the concept of a vector space as such appeared after his work. $\endgroup$ – Margaret Friedland Aug 19 '16 at 18:06

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