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.