# Linear Algebra: Vector Space Over a Field Operation Question

I am confused how the operation $$\cdot : F\times V\rightarrow V$$ is defined for a vector space over a field. Is $$\cdot : F\times V\rightarrow V$$ simultaneously defined as $$\cdot : F\times F\rightarrow F$$ in regards to the field, or should there be symbols distinguishing the axioms below? Any help would be greatly appreciated. If I was to guess the operations are simultaneously defined, but I could be wrong.

Definition 1: A vector space over a field F is defined to be a set $$V$$. This set is said to have two operations $$+: V\times V\rightarrow V$$ and $$\cdot: F\times V\rightarrow V$$. Also, $$V$$ is defined to have the following conditions:

$$(C1)$$ $$V, +$$ forms a commutative group with an identity element $$=0$$,

$$(C2)$$ For all $$a, b\in F$$, for all $$v\in V$$, $$(a⋅ b)⋅ v=a⋅ (b⋅ v)$$,

$$(C3)$$ For all $$v\in V$$, $$1⋅v=v$$,

$$(C4)$$ For all $$a\in F$$, for all $$u, v\in V$$, $$a⋅(u+v)=a⋅u+a⋅v$$,

$$(C5)$$ For all $$a, b\in F$$, for all $$v\in F$$, $$(a+b)⋅v=a⋅v+b⋅v$$.

Definition 2: A $$\textbf{field}$$, $$F$$, is a set with two operations, $$+$$ and $$\cdot$$ which satisfies the following conditions:

$$(C1)$$ $$F, +$$ forms a commutative group with an identity element $$= 0$$,

$$(C2)$$ $$F^{\neq 0}, \cdot$$ forms a commutative group with an identity element $$= 1$$,

$$(C3)$$ For every $$a, b, c\in F$$, $$a\cdot (b+c)=a\cdot b+a\cdot c$$.

These are two distinct operations, for which we use the same symbol. It is usually easy to distinguish between them: if we see $\lambda.v$, then
• if $\lambda\in F$ and $v\in V$, then we are talking about the function from $F\times V$ into $V$;
• if $\lambda,v\in F$, we are talking about the product defined in $F$.
Of course, if it turns out that $V=F$, there is space for ambiguity here, but in this case both operations are usually the same.