TITLE: Geometric interpretation of $\mathbb R^n$
QUESTION [8 upvotes]: Motivated by this question Basis of a basis I've been thinking when we say that a vector can have a geometrical interpretation, are we talking about the vectors themselves or the vectors coordinates in the usual basis $\{(1,0),(0,1)\}$ in for example $\mathbb R^2$?(see picture below to $\mathbb R^2$ case) I think vectors of $\mathbb R^n$ are just n-tuples of real numbers and then don't have a geometrical interpretation, am I right? I'm asking this silly question, because there are a lot of abuse of notation in this part of linear algebra and I want to know the correct concepts in order to not get lost in the future.

Thanks in advance

REPLY [2 votes]: Vectors are abstract elements of a vector space, in fact a vector space is a set of elements called "vectors" with some nice properties.
Note that you can always do formal operations with vectors without using coordinates and only the vector space axioms:
$ v,w \in V(K) $ vector space on $K$ field. $$ v+w = (v+w) \in V(K) $$
the parenthesis are here only to help you think about $v+w$ as an unique element of $V(K)$. 
But if you want to represent this new vector and work with it in the usual manner you have to choose a basis of your space, and in some cases (the cases I think you are interested in, so finite dimensional spaces) use the isomorphism between your space with that basis and $\mathbb{R}^n$ for some $n \in \mathbb{N}$. 
In this way we obtain a representation of a vector by it's coordinates and work with them like they are vectors themselves. Note that if you change the basis you change the isomorphism as well and therefore your coordinates.