I have a vector data set in component-form in two columns with error (x-component and y-component: $x+\delta x$ and $y+\delta y$). I need to rotate this data by an angle $\theta$ to a new coordinate system $x'$ and $y'$.
To do this with perfect data, I understand that you can use the 2D rotation matrix:
$$\begin{bmatrix}x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos (\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}.$$
However, for imprecise data where I have columns for $\delta x$ and $\delta y$, how can I handle this error?
Is it as simple as:
$$\begin{bmatrix}\delta x' \\ \delta y' \end{bmatrix} = \begin{bmatrix} \cos (\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)\end{bmatrix}\begin{bmatrix}\delta x \\ \delta y \end{bmatrix}$$ or something more complicated?