I am given a $3\times 3$ rotation matrix and a $1\times 3$ translation matrix defined in a coordinate system where $x$ points right, $y$ points forwards, and $z$ points up.
Now I need to describe this same rotation and translation in a coordinate system where $x$ points left, $y$ points up, and $z$ points backwards.
I'm not really sure how to go about this. For translation, it seems simple enough to map like: $f(\langle x, y, z\rangle ) =\langle -x, z, -y\rangle$. It's been a while since my linear algebra days, so I'm sure I'm missing something straightforward!
Well, $\begin{bmatrix}-1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix} =\begin{bmatrix}-x\\ z\\ -y\end{bmatrix}$
Call that transformation matrix : $C$.
You have rotation $A$ and translation $b$, such that $ \vec x' = A\vec x+b$, and wish to express $\vec u'$ in terms of $\vec u$ when $\vec u'=C\vec x'$ and $\vec u=C\vec x$.
$$\vec u~{=~C\vec x'\\ ~\vdots~}$$