from: sciencedirect/The axiomatization of linear algebra: 18751940 HISTORIA MATHEMATICA 22 (1995), 262303
When did mathematicians first generally accept the definition of a vector space as
This abstract description did not come into early use, but has the evident advantages of conceptual simplicity and geometric invariance.
Modern linear algebra is based on
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1888 Dedekind expressed best of all how axiomatization played a fruitful role:
The greatest and most fruitful advances in mathematics and the other sciences have been achieved, above all, through the creation and introduction of new concepts, after the frequent recurrence of complex phenomena, which were only laboriously mastered by the old concepts, has forced them upon us. [28, preface]
Essays in the theory of numbers, 1. Continuity of irrational numbers, 2. The nature and meaning of numbers.  Dedekind
1951 Weyl, who felt considerable ambivalence about axiomatization, expressed very well the new role that it was playing:
"Whereas the axiomatic method was formerly used merely for the purpose of elucidating the foundations on which we build, it has now become a tool for concrete mathematical research".
L'idea di vettore come segmento orientato, o bipunto, o differenza di 2 punti 

1827  Barycentric calculus. Möbius (17901868) 
1832  Calcolo delle equipollenze. Giusto Bellavitis (18031880). 
1844  Grassmann calculus of extension, 
1845  "vettore" e' il termine usato per la prima volta da Hamilton, che lo usò per distinguere nei quaternioni la parte vettoriale da quella scalare 
The abstract notion of vector space 

1888  was first isolated by Peano in
geometry. Calcolo geometrico. A
more general concept of vector abstract vector spaces first arose
under a different name, that of "linear system", in the work of Peano. It was not influential then, 
1918  nor when Weyl rediscovered it. Finite dimensional spaces, normed. 
1920  it was rediscovered again by three analysts Banach, Hahn, and Wiener and an algebraist, Noether. Infinite dimensional spaces. 
1941  The notion of vector space over a field defined in an abstract way, in the textbook A Survey of Modern Algebra by the young algebraists Garrett Birkhoff and Saunders Mac Lane, at Harvard, extremely influential in the United States. 
The explicit use of vectors occurs even earlier in the work of
1776 appears in the celebrated Encyclopédie, edited by Diderot, in the article "Rayon vecteur" by the astronomer J.J. de la Lande. He wrote that a radius vector is the "ligne droite qui va ... du soleil au centre de la planète; on l'appelle vecteur, parce qu'on le conçoit comme portant la planète à une de ses extramités .... What de la Lande had in mind becomes clearer when we recall that "vecteur," or "vector," comes from the past participle "vectus" of the Latin verb "vĕho" meaning "to carry or transport.".
1826 in Ampère's
Théorie rnathématique des
phénoménes électrodynamiques and in French mathematical physics.
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Banach spaces complete normed vector spaces.
Banach's paper of 1922, which was his doctoral dissertation of 1920, introduced
the notion of Banach space:
The aim of the present work is to establish certain theorems valid in different functional domains, which I will specify in what follows. Nevertheless, in order not to have to prove them for each particular domain, which would be painful, I have chosen to take a different route ... : I consider sets of elements about which I postulate certain properties; I deduce from them certain theorems, and I then prove for each particular functional domain that the postulates adopted are true for it.
Here Banach used the axiomatic method, but in a way quite different from
Hilbert. Banach's aim was not to characterize a certain mathematical domain by
axioms, as Hilbert had done for Euclidean geometry, but to establish theorems
true for a class of domains by giving axioms for that class, and then to show
that those
theorems were true for a particular domain by showing merely that it satisfied
the axioms. This version of the axiomatic method, so common in mathematics now,
was relatively new in 1920 and not common in analysis, having been used rarely
except in certain branches of algebra (primarily group theory and field theory).
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Thus far we have discussed the geometric and analytic origins of abstract
vector
spaces. Now we turn to the algebraic roots of the modern concept of module.
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1921 The modern concepts of ring, of ideal, and of module over a ring all appeared for the first time in Emmy Noether's groundbreaking paper "Idealtheorie in Ringbereichen".
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Mac Lane took a very positive view of modules in 1981 when
discussing the history of modern algebra:
The first real recognition of the central role of a module is the Bourbaki volume on linear algebra (1947)
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Mac Lane
The formal definition of a vector space by axioms and not by ntuples ... could have been introduced and understood by Grassmann in 1842, it was introduced by Peano in 1888, but it was not introduced and effectively advertised before Weyl (1918) and Banach (1922) .... In the conceptual parts of mathematics, it is not the discovery but the courage and conviction of importance that plays a central role.