The uniqueness of homogeneous matrices of some geometric transformations Given the description of a specific geometric transformation, the homogeneous square matrix of it can be obtained per the methods described by many textbooks. 
For example,  many text books suggest the homogeneous matrix of a reflection be constructed by multiplying a series translations, rotations and so on to the reflection about some axis plane, which can be obtained immediately easily.
Specifically, the homogeneous matrix reflection about plane $x+2y+z+5=0$ can be obtained as:
$$-\dfrac{1}{3}\left[
\begin{array}{cccc}
 -2 & 2 & 1 & 5 \\
 2 & 1 & 2 & 10 \\
 1 & 2 & -2 & 5 \\
 0 & 0 & 0 & -3 \\
\end{array}
\right]$$
and more generally, a reflection about the plane $a X+b Y+c Z+d=0$ can be written into :
$$\left[
\begin{array}{cccc}
 -a^2+b^2+c^2 & -2 a b & -2 a c & -2 a d \\
 -2 a b & a^2-b^2+c^2 & -2 b c & -2 b d \\
 -2 a c & -2 b c & a^2+b^2-c^2 & -2 c d \\
 0 & 0 & 0 & a^2+b^2+c^2 \\
\end{array}
\right]$$
though the construction process for a specific transformation generally is not unique (not only with the non-unique origin reflection about axial reflection, but non-unique in matrix factors' selection and consequences), and of course it is not the definition of the transformation, it   actually proves the existence of the specific reflection's homogeneous matrix. But how can we prove the uniqueness?
I have similar puzzles, i.e., the proof of the uniqueness of their homogeneous matrices,  on such geometric transformations as translation, central projection, rotation, shear, reflection, and so on.
 A: As long as the origin is fixed, these are all ordinary linear transformations, so they are uniquely determined by their matrix, as any linear map $\varphi:\Bbb R^n\to\Bbb R^m$ is just a (left) multiplication by the matrix $M=[\varphi(e_1)\,|\dots|\,\varphi(e_n)]$ where $e_1,\dots,e_n$ is the standard basis of $\Bbb R^n$.
So that, $M\cdot v=\varphi(v)$ for every $v\in\Bbb R^n$.
[Observe that, for any matrix, $M\cdot e_i$ gives the $i$th column of $M$.]
Conversely, if $M\cdot v=\varphi(v)$ for all $v$, then in particular, it applies to the standard basis vectors $e_i$, which shows that the $i$th column of $M$ must be $\varphi(e_i)$, thus proving uniqueness.
If the origin is not fixed, we talk about affine transformations and bring in one more coordinate and the projective plane/space.
Specifically, we embed $\Bbb R^n$ into $\Bbb R^{n+1}$ at the affine hyperplane $H:=\{(x_1,\dots,x_n,1):x_i\in\Bbb R\}$, and apply the transformation $H\to H$ by extending it to a linear transformation $\Bbb R^{n+1}\to\Bbb R^{n+1}$. 
This extension is only unique up to a scalar multiple.
See also https://en.wikipedia.org/wiki/Transformation_matrix#Affine_transformations.
