TITLE: A vector space is an abelian group with some extra structure?
QUESTION [20 upvotes]: Question: A vector space is an abelian group with some extra structure. Given
two vector spaces V1 × V2, show that the group V1 × V2 is a vector space.
Can someone explain to me the first sentence? Especially why it is commutative? As to the second sentence, is it just saying the product of vector spaces is vector space? Why it mentions "group" V1 × V2 particularly?
Thx in advance~

REPLY [12 votes]: "A vector space is an abelian group with some extra structure." This follows from the axioms of a vector space. There are eight of them and the first four ones say exactly that $V$ is an abelian group:

$\forall u,v,w\in V:\ u+(v+w)=(u+v)+w$,
$\forall u\in V:\ u+0=u$,
$\forall u\in V\ \exists -u\in V:\ u+(-u)=0$,
$\forall u,v\in V:\ u+v=v+u$

1-3 say that $V$ is a group, 4 says it is abelian. Now what about the rest of axioms? They concern the multiplication by scalars and they are as follows:

$\forall a\in\mathbb{K}\ \forall u,v\in V:\ a\cdot(u+v)=(a\cdot u)+(a\cdot v)$,
$\forall a,b\in\mathbb{K}\ \forall u\in V:\ (a+b)\cdot u=(a\cdot u)+(b\cdot u)$,
$\forall a,b\in\mathbb{K}\ \forall u\in V:\ a\cdot(b\cdot u)=(ab)\cdot u$,
$\forall u\in V:\ 1_\mathbb{K}\cdot u=u$.

($1_\mathbb{K}$ is the multiplicative neutral element of the field $\mathbb{K}$). What do those axioms say? Let $End(V)$ be the ring of all endomorphisms of $V$ (ie. all group homomorphisms of $V$ into $V$). The multiplication in $End(V)$ is just a composition of functions and its neutral element is the identity function $id_V$: $id_V(u)=u$. Now what does it have to do with Axioms 5-8? They define a homomorphism from $\mathbb{K}$ into $End(V)$! How? Let $\varphi:\mathbb{K}\to End(V)$ be defined as follows $\varphi(a)(u)=a\cdot u$ where $a\in\mathbb{K}$ and $u\in V$. For each $a\in\mathbb{K}$ $\varphi(a)$ is a function from $V$ into $V$ -- it takes an element of $V$ and multiplies it by the scalar $a$. Axiom 5 say that for every $a\in\mathbb{K}$ $\varphi(a)$ is an endomorphism of $V$ ($\varphi(a)\in End(V)$), ie. $\varphi(a)$ is linear. Axioms 6-8 say that $\varphi$ is a homomorphisms between rings $\mathbb{K}$ (every field is a ring with multiplicative identity) and $End(V)$. This homomorphism $\varphi$ is the additional structure you are asking about.