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We know that the reason why we want to introduce a vector space and work with a vector space is that we want to work with a set whose elements can be added and scaled (or a set whose elements are closed under addition and scalar multiplication). Given this motivation, I'm confused by the definition of a vector space:

Definition. A vector Space over field $F$ is a set $V$ such that:

(i). Two operations are defined: vector addition: $V × V → V$ ; scalar multiplication: $F × V → V$ .

(ii). The set V and these two operations satisfy 8 axioms.

Note that as addition is defined as a mapping $V\times V\rightarrow V$, and scalar multiplication is defined as mapping $F\times V\rightarrow V$, this means part (i) of the definition already gives us a set that is closed under finite addition and scalar multiplication.

My question: Given that what we want from a vector space is merely that it is a set whose elements can be added and scaled, what is the primary motivation for imposing the 8 axioms (part (ii) of the definition)?

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    $\begingroup$ Because we want addition and scalar multiplication to behave like addition and scalar multiplication. $\endgroup$ Commented Apr 17, 2020 at 16:04
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    $\begingroup$ See this question and its answers for why closure of the two operations is usually included among the axioms. $\endgroup$ Commented Apr 17, 2020 at 17:30

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Though we gave a preliminary name 'vector addition' (and a suggestive notation '$+$') to the operation $V\times V\to V$, it is not assumed that it is indeed 'an addition' operation on some known structures.
In itself it can be any two variable function on $V$.

Instead, we assume the basic and most important properties (the axioms) to try to capture what it means to be an 'addition-like operation'.

Similarly for the scalar multiplication.

Note also that $F$ is already assumed to be a field, i.e. to be equipped with (constants named $0$ and $1$), an addition(-like operation), a substraction(-like operation), a multiplication(-like operation), and a division(-like operation) by any nonzero element.

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  • $\begingroup$ Thanks a lot. This is very helpful! $\endgroup$ Commented Apr 18, 2020 at 1:22
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Without the axioms of addition and scalar multiplication, we wouldn't know addition was commutative. Let $v,w \in V$ a space with two operations $+ : V \times V \to V$ and $\cdot: F \times V \to V$. To build up the theory of vector space, we need to be able to say things like $v+w = w+v$. But this is only true if we require the axiom of commutativity to hold for the $+$ operation. Similarly we want distribution of scalar multiplication $c(v+w) = cv+cw$. Again this only holds if we enforce axioms.

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  • $\begingroup$ Thanks! This is very helpful. $\endgroup$ Commented Apr 18, 2020 at 1:21

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