How to write operators in matrix notation I'm trying to figure out how to do this, I've checked 
here and what I got from it is
Suppose you have an operator $D \equiv \frac{d}{dx}$ and some random basis $\vec{e} = \{ cos (ax), sin (bx) \} $
Then the operator $D$ would be represented in matrix notation as
$$D = \begin{pmatrix}0
 &b \\ -a
 & 0
\end{pmatrix}$$
Is this it?
So it is safe to say the operator will always be a $n \times n$ matrix, where $n$ is the vector space's dimension?

 A: Let $V$ be the subspace of smooth functions spanned by 
\begin{align*}
e_1(t) &= \cos(at) & e_2(t)=\sin(bt)
\end{align*}
Let $W$ be the subspace of smooth functions spanned by
\begin{align*}
f_1(t) &= \cos(bt) & f_2(t)=\sin(at)
\end{align*}
Let $d:V\to W$ be the linear operator defined by $df(t)=f^\prime(t)$. Note that
\begin{array}{rcrcrcrc}
de_1(t) & = & -a\sin(at) &=& \color{red}{0}\, f_1(t) &+& (\color{red}{-a})\,f_2(t) \\
de_2(t) &=& b\cos(bt) &=& \color{blue}{b}\,f_1(t) &+& \color{blue}{0}\,f_2(t)
\end{array}
This implies that the matrix representing $d$ in the given bases is
$$
D=
\begin{bmatrix}
\color{red}{0} & \color{blue}{b}\\
\color{red}{-a} & \color{blue}{0}
\end{bmatrix}
$$
as expected.

Here's the general situation:
Let $T:V\to W$ be a linear map and let 
\begin{align*}
\alpha &= \{v_1,\dotsc,v_n\} &
\beta  &= \{w_1,\dotsc,w_m\}
\end{align*}
be bases for $V$ and $W$ respectively. This means there are unique scalars $\lambda_{ij}$ for $1\leq i\leq m$ and $1\leq j\leq n$ such that
$$
\begin{array}{ccccccccc}
T(v_1) & =      & \color{red}{\lambda_{11}}w_1 & + & \color{red}{\lambda_{21}}w_2 & + &\dotsb &+ & \color{red}{\lambda_{m1}}w_m \\
T(v_2) & =      & \color{blue}{\lambda_{12}}w_1 & + & \color{blue}{\lambda_{22}}w_2 & + &\dotsb &+ & \color{blue}{\lambda_{m2}}w_m \\
\vdots & \vdots & \vdots          &\vdots &\vdots       &\vdots &\ddots&\vdots&\vdots \\
T(v_n) & =      & \color{green}{\lambda_{1n}}w_1 & + & \color{green}{\lambda_{2n}}w_2 & + &\dotsb &+ & \color{green}{\lambda_{mn}}w_m \\
\end{array}
$$
The matrix of $T$ relative to the bases $\alpha$ and $\beta$ is defined as
$$
[T]_\alpha^\beta=
\begin{bmatrix}
\color{red}{\lambda_{11}} & \color{blue}{\lambda_{12}} & \dotsc & \color{green}{\lambda_{1n}} \\
\color{red}{\lambda_{21}} & \color{blue}{\lambda_{22}} & \dotsc & \color{green}{\lambda_{2n}} \\
\vdots       & \vdots       & \ddots & \vdots \\
\color{red}{\lambda_{m1}} & \color{blue}{\lambda_{m2}} & \dotsc & \color{green}{\lambda_{mn}}
\end{bmatrix}
$$
Note that this is an $m\times n$ matrix.
