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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 $\mathbb R^n$.2012-12-01
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    A linear space is a space where it makes sense to form linear combinations.2012-12-01
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    @Hans: That sounds suspiciously circular. What are linear combinations, then? Things which we take in linear spaces?2012-12-01
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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 $x^2 e^{y}$.2012-12-01
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    @AsafKaragila if that sounds circular what about vector spaces? As is, what are vectors? The things that form a vector space.2015-07-01
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    FWIW, (the Russian equivalent of) “linear space” is indeed more frequently used in Russian than “vector space”.2017-11-19

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