Mistake on Wikipedia about coordinate change? I'm reading this Wikipedia article about coordinate change.
Say, $\alpha_i$ is some basis of some finite vector space. Then a vector $\xi$ can be written as a coordinate tuple with respect to this basis:
$$ \xi = \sum_i x_i \alpha_i$$
Say $M$ is a matrix that changes the basis, say, $M \alpha_i = \alpha_i'$. Now Wikipedia states:
This means that given a matrix $M$ whose columns are the vectors of the new basis of the space (new basis matrix), the new coordinates for a column vector $v$ are given by the matrix product $M^{−1}v$. 
So far so good: If $M \alpha_i = \alpha_i'$ then $ \alpha_i =M^{-1} \alpha_i'$ and therefore
$$ \xi = \sum_i \alpha_i x_i = \sum_i M^{-1}\alpha_i' x_i$$
But now I'm confused: $\xi$ should be a basis independent entity. Only its coordinates should change if the basis changes. But applying $M$ to the equality above we get
$$ M \xi = \sum_i \alpha_i' x_i$$
So when we change the basis the coordinates of $\xi$ (the $x_i$) stay the same but the left hand side changes instead. 

Please could someone help me resolve my confusion?

 A: Let there be given two bases $\{\alpha_1, \ldots, \alpha_n\}$ and $\{\alpha'_1, \ldots, \alpha'_n\}$ yielding coordinates $(x_1,\ldots, x_n)$ and $(x'_1,\ldots, x'_n)$ respectively for a particular vector $\xi$.
There is indeed a linear transformation $M$ (not really a matrix) such that $M\alpha_i=\alpha'_i$ and $\alpha_i=M^{-1}\alpha'_i$ for $i\in \{1,\ldots, n\}$.
The equations 
$$\sum x_i\alpha_i=\xi=\sum x'_i\alpha'_i=\sum x'_iM^{-1}\alpha_i$$
and 
$$\sum x_i M\alpha_i=M\xi=\sum x'_i\alpha_i$$ are both completely valid.
There isn't any contradiction here: $M\xi$ is a completely new vector with coordinates $(x'_1,\ldots, x'_n)$ in the $\alpha$ basis, which is, of course, a completely different thing from $\xi$ which has those same coordinates, but with respect to the $\alpha'$ basis.
I think the thing you're running up against is the difference between active and passive roles of a linear transformation. By writing $M\xi$ you're probably thinking of an active transformation (it leaves the coordinates alone but moves the basis elements, which will typically move vectors) but you are trying to reconcile that with a coordinate transformation (a passive transformation: the coordinates are changing but the vectors are not) which we think of as moving the coordinates and not the vectors themselves.
When you're dealing with a passive transformation, you typically want to think about a matrix acting on the list of coefficients, and when you want to think about an active transformation you think about a linear transformation acting on abstract vectors.
