The question is quite interesting, at least from an epistemological point of view. It is not easy to work out an all-comprehensive answer which is in the meantime meaningful enough in a broad context, but I will try to say something that hopefully can be understood, at least on a metamathematical level, whatever this may mean. Also, since you have tagged the question under "Category Theory", I will feel free to use that language, which, unsurprisingly, provides quite an enlightening way of interpreting the problem.
So, the main difference between an "internal" and an "external" approach may be summarised as follows. Suppose you have fixed a mathematical entity (or structure) $\mathcal{M}$ which you are interested in studying, for example a vector space $V$. For our purposes, I guess one should think to such a mathematical entity as a mathematically definable object, characterised by some defining properties (of course, this "definition" is deliberately vague).
Investigating the nature of $\mathcal{M}$ internally means to study $\mathcal{M}$ using solely the structure that $\mathcal{M}$ has in its own and the properties that the axioms defining $\mathcal{M}$ imply, as if $\mathcal{M}$ had suddenly become the only available working ground, a comprehensive universe into which doing your math. In particular, one is not allowed to refer to any other entity besides $\mathcal{M}$ or besides those that can be defined or built up in $\mathcal{M}$. For example, if $\mathcal{M}$ is your favourite vector space $V$, looking at its linear subspaces is an internal way of proceeding, because subspaces are substructures of your given entity. You can then do some operations on subspaces, such as taking intersections or internal direct sums, which are internal as long as they can be performed inside $V$ and their result still lies inside $V$.
On the other hand, an external study of $\mathcal{M}$ would try to grasp information about $\mathcal{M}$ basing on the connections that $\mathcal{M}$ may have with other entities, or, better, seeing $\mathcal{M}$ as within (or as related to) a wider landscape, a working universe to which one is allowed to link $\mathcal{M}$. For instance, once you have taken your favourite vector space $V$ and a bunch of subspaces $V_{i}$ of $V$, you may realise that such subspaces are themselves vector spaces and constitute mathematical entities on their own right, even if of a kind similar to the one of your $V$. In other words, both $V$ and the $V_{i}$ are objects of the category of vector spaces (over some field). Then, you may perform some operations on these $V_{i}$ seeing them as structures on their own, and not as substructures or $V$ (i.e. considering them as objects in the category of vector spaces and not as subobjects of $V$). For example, you may take the external direct sum, or the direct product of the $V_{i}$ and then, you may wonder if the results of such operations can be related to your $V$, giving some informations on it ($V$ may be isomorphic to the external direct sum of the $V_{i}$ or a quotient of them).
The best way to explain the two approaches comes thinking at Category Theory. Take a (locally small) category $\mathcal{C}$ (even better, an elementary topos). Then such a category is naturally related to the category $\mathbf{Set}$ of sets through the Hom functors. Now, one can study properties of $\mathcal{C}$ or even define some notions internally to $\mathcal{C}$, using only the given structure of $\mathcal{C}$, i.e. its objects, its arrows and its composition law, or externally transferring the properties and the definitions to the category of sets.
For instance, given objects $X,Y$ of $\mathcal{C}$, one can define their product $X\times Y$ either internally, requiring that $X\times Y$ is an object of $\mathcal{C}$ equipped with two morphisms in $\mathcal{C}$ satisfying the well-known universal property, or externally, requiring that the functor
$$
\mathcal{C}(-, X)\times \mathcal{C}(-,Y)
$$
is representable, i.e. requiring that, for all $Z\in\mathcal{C}$, $\mathcal{C}(Z,X\times Y)$ is a product in the category of sets. The same kind of double approach can be carried over to loads of notions commonly used in Category Theory, such as limits, colimits, subobject classifiers, exponentials etc.
It turns out that (at least in the abovementioned examples) the two approaches are equivalent, in the sense that an object (together with a family of arrows) in $\mathcal{C}$ satisfy a property internally if and only if it satisfies it externally.
However, the two ways of investigating the category $\mathcal{C}$ are sensibly different: the internal one makes assertions just about the structure of $\mathcal{C}$, without any further (set-theoretic) assumptions and can therefore be used to give a self-referring description of $\mathcal{C}$, as if it was the universe into which we set our play. On the other hand, the external point of view studies $\mathcal{C}$ as a structure constructed within a working, fundational environment (some kind of set-theory), into which (and dipendently to which) one develops the whole theory on $\mathcal{C}$.
The internal and the external point of view are a common feature in Category Theory (hence, in the whole Mathematics) and their interplay and connections are not only what makes the subject delightful but provide deep results (just to mention an example that springs to my mind, think about Giraud's Theorem characterizing axiomately a Grothendieck topos).
Best Answer
Addition is a binary operation: it takes two vectors, and returns a vector. By induction, we can add finitely many vectors together. But we cannot add infinitely many vectors together.
So, for example, the vector $(1,1,1,1,\ldots)$ cannot be expressed as a linear combination of the vectors $\mathbf{e}_j$ (where $\mathbf{e}_j$ is an element of $\mathbb{R}^{\omega}$ in which the $i$th coordinate of $\mathbf{e}_j$ is $1$ if $i=j$ and $0$ otherwise), because by definition, a linear combination has only finitely many summands. So a linear combination of the $\mathbf{e}_i$ has only finitely many nonzero entries.
If you take the collection of vectors you propose, a linear combination of them, since by definition it can only involve finitely many vectors from your collection, will necessarily have zero entry in all but finitely many coordinates.