Finding the matrix representation of a certain function

So I need to find the matrix representation of the function f.

f = [.]β given by [$\vec v$]β = $\begin{pmatrix}a_1\\a_2\\a_3\\a_4\end{pmatrix}$ where $\vec v = a_1\vec v_1 + a_2\vec v_2 + a_3\vec v_3 + a_4\vec v_4$.

The basis β = {$\begin{pmatrix}1\\1\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\1\\0\end{pmatrix}, \begin{pmatrix}0\\0\\1\\1\end{pmatrix}, \begin{pmatrix}0\\0\\0\\1\end{pmatrix}$} = {$\vec v_1, \vec v_2, \vec v_3, \vec v_4$}.

The problem is that I don't know what f = [.]β means. I was told that its the function that takes as input a vector $\vec v$, and outputs the column vector $a_1 a_2 a_3 a_4$ otherwise known as the "coordinates with respect to the basis beta" function but I'm still not sure what I need to do.

• Well firstly, you need some prescription for the function $f$. – user137731 Nov 22 '16 at 21:56
• Then you can look at my answers to these questions to try to understand what matrix representation of a function means: 1, 2, 3, 4. – user137731 Nov 22 '16 at 22:03
• I do know what the matrix representation of a function is, I just don't know how to find the specific function. – david mah Nov 22 '16 at 23:09
• That conflicts with the first sentence of your question. But OK. You still need some prescription for $f$ -- or its action on a basis or something tangible about it. Is it supposed to be the change of basis matrix from the standard basis to $\beta$? – user137731 Nov 22 '16 at 23:12
• It is a change of basis matrix from the standard basis to β. I know how to find the change of basis but what I have no idea how to do is with that dot. All it says is that f is some dot with respect to the basis but I have no idea how to do that. – david mah Nov 22 '16 at 23:18