<s>[INST] Isometry and its inverse I got this affine map:
$$
f: R^3  \rightarrow R^3: \begin{pmatrix}x\\
y\\
z\\\end{pmatrix} \rightarrow A \cdot \begin{pmatrix}x\\
y\\
z\\\end{pmatrix} + \begin{pmatrix}0\\
-1\\
1\\\end{pmatrix}
$$
with $$ A = \begin{bmatrix}
1 & a_{12} & a_{22}\\
0 & 1 & a_{21}\\
0 & 0 & 1
\end{bmatrix}$$
Also given was this information about the inverse (which should also be an affine map?):
$$
g=f^{-1}: R^3  \rightarrow R^3: \begin{pmatrix}x\\
y\\
z\\\end{pmatrix} \rightarrow B \cdot \begin{pmatrix}x\\
y\\
z\\\end{pmatrix} + \bar b
$$
I have to find $\bar b$. Does anyone have an idea how to find it? I tried inputting some values but I can't seem to get there.
 [/INST] $\overline{b} = g(0) = f^{-1}(0)$, so solve $f(x) = 0$ so $Ax = (0,1,-1)$. The answer (which depends on the 3 constants in $A$) is the required $\overline{b}$.
</s>