# Why we need the "8 axioms of addition and multiplication" in the definition of a vector space?

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)?

• Because we want addition and scalar multiplication to behave like addition and scalar multiplication.
– user658409
Apr 17, 2020 at 16:04
• See this question and its answers for why closure of the two operations is usually included among the axioms.
– amd
Apr 17, 2020 at 17:30

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.

• Thanks a lot. This is very helpful! Apr 18, 2020 at 1:22

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.

• Thanks! This is very helpful. Apr 18, 2020 at 1:21