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I have never come across the term 'linear space' as a synonym for 'vector space' and it seems from the book I am using (Linear Algebra by Kostrikin and Manin) that the term linear space is more familiar to the authors as opposed to using vector space. This book was translated from the Russian edition into English so it seems that the term linear space is/was more predominant in the Russian speaking countries?

So I was wondering what is the intuition/motivation behind choosing such a term for the concept of a vector space. Why have the word 'linear space' for vector spaces? What is so "linear" about vector spaces? Is it possible to have a "non-linear" vector space? Why should we distinguish between "linear" and "non-linear" if such a term non-linear space exists?

I know that I have not had enough linear algebra and exposure to higher mathematics to have a feel for why such a term is used for vector spaces and it would be great if someone could give an exposition.

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Because vectors are "lines" if you think about them in $\mathbb R^n$. –  Asaf Karagila Dec 1 '12 at 10:54
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A linear space is a space where it makes sense to form linear combinations. –  Hans Lundmark Dec 1 '12 at 10:56
    
@Hans: That sounds suspiciously circular. What are linear combinations, then? Things which we take in linear spaces? –  Asaf Karagila Dec 1 '12 at 11:19
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@Asaf: Something like that, yes. :-) But seriously, once we have come to associate the word "linear" with first degree polynomials, I don't think it's very far-fetched to call the expression $3x+2y$ a "linear" combination of the quantities $x$ and $y$, as opposed to some more arbitrary combination like $x^2 e^{y}$. –  Hans Lundmark Dec 1 '12 at 15:20
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1 Answer

I think the following article:

Gregory H. Moore. The axiomatization of linear algebra: 1875-1940. Historia Mathematica, Volume 22, Issue 3, 1995, Pages 262–303

(Available here from Elsevier) may shed some light on your question, although you may not have enough mathematical experience to understand the entire article. Here is my understanding having browsed the article, but I must stress that I am not a mathematical historian, so please don't quote me!

The idea of an abstract space where an addition is defined between elements and there is a field action (rather than a particular realization as, for instance, $\mathbb{R}^n$ or $C([0,1])$) seems to be due to Peano in 1888, where he called them linear systems. The definition of an abstract vector space didn't catch on until the 1920s in the work of Banach, Hahn, and Wiener, each working separately. Hahn defined linear spaces in order to unify the theory of singular integrals and Schur's linear transformations of series (both employing infinite dimensional spaces). Wiener introduced vector systems which seems to be roughly equivalent to Banach's definition, which was motivated by finding a common framework to understand integral operators (Banach's 1922 paper "Sur les operations dans les ensembles abstraites et leur application aux équations intégrales" is available online and is quite readable) which were defined on champs (domains).

I understand the modern name vector space is popular because of a widely circulated 1941 textbook by Birkhoff and MacLane, A Survey of Modern Algebra, where the term is used.

As Asaf and Hans have indicated in their comments, the motivation for calling such spaces vector spaces is because intuitively, they generalize our understanding of "vectors" (differences between points) in a finite dimensional Euclidean. The motivation for calling such spaces linear spaces is because our ability to add together different elements is the crucial feature which lets us apply the general theory to solve specific problems which are not obviously (to the 1920's eye) about vectors (in particular, in PDE and mathematical physics).

In your course, it is unlikely you will cover material that requires this abstraction, but it is a good habit for later mathematics to work in generality while you maintain your intuition in concrete examples.

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