Let us say I have a 'vector' $\vec v$ for which I can do the following operation on $A\vec v$ where $A$ is a matrix. Now most people (i think) would say that $\vec v \in R^n$ however $\vec v$ is not a an $n$-tuple, it can't be since matrices can't act on tuples and therefore $\vec v$ is not in $R^n$. Does this mean $\vec v$ is not a vector in terms of been an element of a vector space?
(I agree that there is an isomorphism between the vector which you use a matrix on and n-tuples)
The issue here is that the tuple $\vec{x}=(x_1, \dots, x_n)$ is not really different from the coordinates of $\vec{x}$ relative to the canonical basis of $R^n$ formed by the element $e_1 =(1,0, \dots, 0)$, $e_2= (0,1, 0, \dots, 0)$ and so on.
Absent any indication to the contrary one would usually assume that the matirx is meant with respect to the canonical basis and everything is fine.
It is however true that in a strict sense the operation of a matrix on the elements of a vector space only really makes sense when an ordered basis is given (explicitly or implicitly).
There might also be a second issue, if one is very strict, namely, that of rows versus columns. However, it seems this is not the current convern.