<s>[INST] Row and Column rotation matrices Today I started studying Rotation matrices and derived a rotation matrix like this:
\begin{equation*}
\begin{pmatrix}
       x & y
\end{pmatrix}
\begin{pmatrix}
        \cos(\beta) & \sin(\beta)  \\
        -\sin(\beta) & \cos(\beta) 
        \end{pmatrix}
\end{equation*}
When I was googling for the solution to check if this was ok, I found almost everybody expressed it like this:
\begin{equation*}
\begin{pmatrix}
        \cos(\beta) & -\sin(\beta)  \\
        \sin(\beta) & \cos(\beta) 
\end{pmatrix}
\begin{pmatrix}
        x \\ y
\end{pmatrix}
\end{equation*}
If I'm not mistaken, this leads to the same equations, but is there anything else special about expressing it in these 2 different forms? Has this something to do with  Handedness?
 [/INST] The difference is that you used a row vector and the other expression uses a column vector. And they are the same if you transpose one of them. 
\begin{align}
\left( 
\begin{pmatrix}
       x & y
\end{pmatrix}
\begin{pmatrix}
        \cos(\beta) & \sin(\beta)  \\
        -\sin(\beta) & \cos(\beta) 
\end{pmatrix}
\right)^T
&=
\begin{pmatrix}
        \cos(\beta) & \sin(\beta)  \\
        -\sin(\beta) & \cos(\beta) 
\end{pmatrix}
^T
\begin{pmatrix}
       x & y
\end{pmatrix}
^T
\\ &=
\begin{pmatrix}
        \cos(\beta) & -\sin(\beta)  \\
        \sin(\beta) & \cos(\beta) 
\end{pmatrix}
\begin{pmatrix}
      x \\ y
\end{pmatrix}
\end{align}
</s>