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.
 A: The vector $(11,-16)$ is written in the standard basis - it means $11\times(1,0)+(-16)\times(0,1)$. This is exactly the computation that you do when you write:
$$2u_1+3u_2=2(1,-2)+3(3,-4)=(11,-16)$$
If we write $e_1=(1,0)$ and $e_2=(0,1)$, then what you're writing is:
$$2u_1+3u_2=2(e_1-2e_2)+3(3e_1-4e_2)=11e_1-16e_2$$
so you've changed coordinates from $u_1$ and $u_2$ to $e_1$ and $e_2$.
It's not strictly correct to say that a vector $(a,b)$ "belongs to" a basis, but you do need to know what basis you're working in to be able to interpret it.
More about vectors vs. coordinates
Normally we take $\mathbb{R}^2:=\{(a,b):a,b\in\mathbb{R}\}$ to be the set of pairs of real numbers, with vector space structure given by:
$$(a,b)+(c,d)=(a+c,b+d)$$
$$\lambda(a,b)=(\lambda a,\lambda b)$$
So its points (vectors) are pairs $(a,b)$. We also have that $(a,b)=a(1,0)+b(0,1)$, so the vectors $e_1=(1,0)$ and $e_1=(0,1)$ span (I'm not claiming they're a basis yet, but of course they are), but I can still understand $(a,b)$ without knowing about $e_1$ and $e_2$.
Now pick a basis $v_1,v_2$ of $\mathbb{R}^2$ (not necessarily the standard one). Then for any $u\in\mathbb{R}^2$, we have $u=xv_1+xv_2$, so we could write $u=(x,y)$ in the coordinates $v_1$ and $v_2$. However, because $u$ is a point of $\mathbb{R}^2$, it is a pair $(a,b)$ of real numbers. But unless $v_1=e_1$ and $v_2=e_2$, we won't have $a=x$ and $b=y$. This is confusing (now we can truthfully say $u=(a,b)$ and $u=(x,y)$, but the pair $(a,b)$ isn't equal to the pair $(x,y)$), so instead I'll write $u=[x,y]_v$ to mean that $u$ has coordinates $x$ and $y$ in the basis $v_1$ and $v_2$. Now we can say $(a,b)=[x,y]_v$ without getting confused, and our notation separates a vector in $\mathbb{R}^2$, which is just a pair of numbers $(a,b)$ that doesn't depend on any basis, from a coordinate representation $[x,y]_v$, which requires the basis $v$ to be understood. (Note that we do have $(a,b)=[a,b]_e$).
Now to return to your example, $u_1$ really is the vector $(1,-2)$, and $u_2$ really is the vector $(3,-4)$ - I don't need any basis to understand this. In our new coordinate-emphasising notation, your calculation is now:
$$[2,3]_u=2u_1+3u_2=2(1,-2)+3(3,-4)=(11,-16)$$
where $(11,-16)$ is interpreted as just being a vector in $\mathbb{R}^2$, with no chosen basis (although we could think of it as $(11,-16)=[11,-16]_e$ if we wanted). Now to write it in terms of the basis $v_1$ and $v_2$ you need to find $x,y$ such that $[x,y]_v=(11,-16)$, or rather such that:
$(11,-16)=[x,y]_v=xv_1+yv_2=x(1,3)+y(3,8)=(x+3y,3x+8y)$
A: When starting out, it's often easy to confuse the basis coordinates and the elements of $\Bbb R^2$ because they are all written as ordered pairs with parentheses. 
Let's try to keep them separate by using $\langle a ,b\rangle$ for coordinates in $S_1$, $[a,b]$ for coordinates in $S_2$ and just $(a,b)$ for elements of $\Bbb R^2$. This should help you envision "which coordinates belong to which basis," and help you keep those separate from the elements of $\Bbb R^2$.
Then $\langle2,3\rangle$ is the coordinates for the vector $2(1,-2)+3(3,-4)=(11,-16)$
To find the coordinates $[a,b]$ for $(11,-16)$, you'll have to solve the following equation:
$$a(1,3)+b(3,8)=(11,-16)$$

Comment: what others have said about interpreting $(a,b)$ as coordinates of "the standard basis" is correct, but I don't find it immediately relevant. I think it's pedagogically more helpful to emphasize their identity as elements of the vector space, rather than yet another set of coordinates. 
We need to get you to recognize that the coordinates of a point in a basis are the coefficients you need to manufacture that vector using the basis :)
A: The basis for everything, unless specified, is the standard basis $\{\textbf{e}_1=(1,0),\textbf{e}_2=(0,1)\}$
