In an exercise I'm asked the following:
Consider $\mathbb R^2$ with an inner product whos metric matrix relative to the standard basis is: $$\left( \begin{matrix} 4 & 1 \\ 1 & 1 \end{matrix} \right)$$ and the basis of $\mathbb R^2$, $\mathcal {B} = ((1/2,0) , (1/\sqrt{12},-4/\sqrt{12}))$.
What is the second coordinate of the vector $(1,\sqrt{12})$ in the basis $\mathcal B$
The thing that's bothering me in this question is: Why do we need to use the metric matrix to find out the coordinates of the vector? Wouldn't just a simple change of basis work, or is there something that I'm missing? Thanks
You are right, a simple basis transformation would do it.
However, you can alternatively aim for a specific $i$th coordinate in an orthonormal basis $e_j$ by simply evaluating $\langle x,e_i\rangle$.
So now if you have verified that the given basis is orthonormal for the given inner product (which it is), then $$\pmatrix{1&\sqrt{12}}\pmatrix{4&1\\1&1}\pmatrix{1/{\sqrt{12}} \\ -4/\sqrt{12}}$$ will give you the second coordinate in $\mathcal B$.