Basis of a basis I'm having troubles to understand the concept of coordinates in Linear Algebra.
Let me give an example:
Consider the following basis of $\mathbb R^2$:
$S_1=\{u_1=(1,-2),u_2=(3,-4)\}$ and $S_2=\{v_1=(1,3),v_2=(3,8)\}$
Let $w=(2,3)$ be a vector with coordinates in $S_1$, then $w=2u_1+3u_2=2(1,-2)+3(3,-4)=(11,-16)$.
When I tried to found the coordinates of $w$ in $S_2$, I found the following problem:
Which basis $(11,-16)$ belongs to? I suppose the same of $u_1$ and $u_2$, but which basis $u_1$ and $u_2$ belongs to? and if I discover the basis of $u_1$ and $u_2$, what's the basis of the basis of $u_1$ and $u_2$?
I found an infinite recurrence problem and I was stuck there.
Maybe I'm seeing things more complicated than it is, but it seems that there is a deeper and philosophical question inside of this doubt, I couldn't see what a coordinate really is.
I would be very grateful if anyone help me with this doubt.

The basis for everything, unless specified, is the standard basis $\{\textbf{e}_1=(1,0),\textbf{e}_2=(0,1)\}$
