What exactly is a "vector-space structure?" (Linear Alg) I am supplementing my linear algebra book with wikipedia, and I came across an interesting term that isnt mentioned precisely by that name in my textbook. 
The full sentence is, "Similarly as in the theory of other algebraic structures, linear algebra studies mappings between vector spaces that preserve the vector-space structure." This section was discussing linear transformation.
My question is, what exactly is a vector-space structure at its most fundamental level and how can I really understand it? I have an okay grasp on the concept of a vector space, however is a structure something that is distinct from a space?
 A: A vector space is defined as a quadruple $(\mathbf{V},\mathbb{K},\oplus,\odot)$ where $\mathbf{V}$ is a set of elements called vectors, $\mathbb{K}$ is a field $(\mathbb{K},+,\cdot)$ , $\oplus$ is a binary operation (called sum) on $\mathbf{V}$ such that $(\mathbf{V},\oplus)$ is an Abelian group and $a\odot\mathbf{v}:\mathbb{K}\times\mathbf{V} \rightarrow \mathbf{V}$ is a scalar multiplication such that, $\forall a,b \in \mathbb{K}$ and $\forall \mathbf{u,v} \in \mathbf{V}$ we have:
$$
a\odot(b\odot\mathbf{v})=(a\cdot b)\odot\mathbf{v}  
$$
$$
1\odot\mathbf{v}=\mathbf{v}
$$
$$
a \odot (\mathbf{u}\oplus\mathbf{v})=a \odot\mathbf{u}\oplus a\odot \mathbf{v}
$$
$$
(a+b)\odot \mathbf{v}=a\odot \mathbf{v}\oplus b\odot \mathbf{v}
$$
Note that $(+,\cdot)$ are the operations on $\mathbb{K}$ and are different from the operations $(\oplus, \odot)$.
This ''structure'' is a generalization of the $3$ dimensional space of geometry, where the vectors are oriented segments and between them we define an ''addition'', using the parallelogram low, and a ''scalar multiplication'' by a real number.
A: The "structure" in this case is just the specification of its algebraic operations.
When one says "this set has a vector space structure" this means that the abelian group and scaling operations that make it into a vector space are given. In other words, we are emphasizing that it satisfies the axioms that make it a vector space under those operations.
Similarly one can say "look at the abelian group $\Bbb Z/p\Bbb Z$ for a prime $p$. It has a ring structure given by ordinary multiplication mod $p$. Additionally, it has a vector space structure using the same operation."
A: By "structure" they're referring to the "algebraic structure" of the set.  Algebraically, a vector space is a module over a field.  You need to learn what those things are in a basic book on "abstract algebra".
A: Think of the structure of integers under addition and multiplication, this is just some rules for some objects. A vector space is similar but much more complicated(atleast at the first look). A set with rules och operations, this is what one would call a "space" in math. It takes a while to get used to the idea.
