Silly question about the meaning of "preserving structure". First of all, I would like to ask to you, to restrict the answers just in the context of Real Vector Spaces.
So, my question is quite simple: what supposed to mean the phrase: "this map preserves the vector space structure. Or, under these defined operations, the vector space structure is preserved".
For instance:
Let F be a field. And V and W vector spaces.
Consider,now, a map:
$T: U \to V$; $u_{1}\boxplus_{U} u_{2} \mapsto T(u_1\boxplus_{U} u_2) := T(u_1)\boxplus_{V} T(u_2) $
Well, what is the difference between the $\boxplus _{U}$ and $\boxplus _{V}$?
And what this difference is supposed to mean an how is related with the phrase? What is the undoubtly right way to interpret these kind of notions everytime?
[My first guess is the thought: "well, under the $\boxplus _{U}$ the $T(u_1\boxplus_{U} u_2)$ is an element of a set V. If this V forms a vector space under the $\boxplus _{V}$ (why occurs the change of $\boxplus _{U}$ and $\boxplus _{V}$?)  ,then this is the meaning of "preserve structure".]
Obs:------------------------------
Is there any sense to ask the difference of: A map that preserve structures and a operation that preserve structures?
 A: A vector space $V$ over a field $F$ is a set together with certain operations such that certain conditions are fulfilled. In particular, we have a binary operation $V\times V\to V$ that we call addition and might denote as $\boxplus_V$ and that makes $V$ an abelian group, and we have an action of the field $F$ on this group, i.e., a map $F\times V\to V$ (which we might want to write $\boxdot_V$). 
If $U$ is another vector space, we have the same kind of structure, but use different symbols $\boxplus_U$, $\boxdot_U$ to denote them. 
Of course, $\boxplus_U$ and $\boxplus _V$ are different maps, already because they are defined on different domains, similar for $\boxdot_U$ and $\boxdot_V$. (Actually, we do usually use the common symbols $+$ and $\cdot$ throughout and hope that no confusion arises, but to cover the most general situation, e.g., $V$ is already endowed with an addition, but we want to us a different operation, or $U$ and $V$ have elements in common, but the sums of these differ, we better be careful with the notation)
So if we pick three vectors $u,v,w\in V$, it may (or may not) be the case that $w$ is the sum of $u$ and $v$. A structure-preserving map $T\colon U\to V$ now is one that preserves the sum relation, i.e., whenever we have $u,v,w$ such that $w$ is the sum of $u$ and $w$ ($w=u \boxplus_V v$, then their images $Tu, Tv, Tw$ are in the same (or rather corresponding) relation ($Tw=Tu\boxplus_U Tv$).
Of course, we also postulate that $T$ respect the scalar multiplication, i.e., for any $\alpha\in F$. whenever  $v,w\in V$ are in the relation that $w$ is thw $\alpha$-multiple of $v$ ($w=\alpha\boxdot_V v$), then so are their images$Tv$ and $Tw$ (i.e.,$Tw=\alpha\boxdot_U Tv$)
A: The simple answer is that what is meant is $T$ is a linear transformation from $U$ to $V$.
In other words:
$T((a\boxdot_U u_1) \boxplus_U (b\boxdot_U u_2)) = (a\boxdot_V T(u_1)) \boxplus_V (b\boxdot_V T(u_2)).$
Usually the operations are understood to be the operations that make $U,V$ vector spaces in the first place, and we write simply:
$T(au_1 + bu_2) = aT(u_1) + bT(u_2)$.
Put another way, linear combinations (over $F$) of elements of $U$ are mapped by $T$ to the same linear combination of the images (under $T$, over $F$). 
So the "vector-space-ness" of $U$ is preserved in the image (range) of $T$ in $V$.
One has to be careful, here: it is usually not the case that we have a "copy" of $U$ inside $V$, because $T$ may not be one-to-one. What typically happens is that $T$ condenses the space $U$ into a (perhaps) smaller subspace of $V$.
