<s>[INST] Finding a linear transformation matrix for a set of vectors I am wondering if we can find a linear transformation matrix $A$ of size $3\times 3$ over the field of two elements $\mathbb{Z}_2$ i.e. a matrix $A$ of zeros and ones s.t.
$$A \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, 
A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, 
A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},
A \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix},
A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix},
A \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix},
A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$$
As can be seen above, the vectors I am looking at are all the possible 3D vectors with zeros and ones. $A$ is supposed to be an invertible transformation such that $A\neq A^{-1}$ because the effect of $A$ is a bijection from the space of all the possible 3D vectors with zeros and ones to that space itself.
I found an invertible $A$ using three of the equations above but that $A$ does not work for other vectors and so am wondering if it is possible to find such an $A$?
 [/INST] There is no such $A$, invertible or not. For instance your equations lead to
$$
\begin{bmatrix} 1 \\ 0\\ 0\end{bmatrix}=\begin{bmatrix} 1 \\ 1 \\ 1\end{bmatrix}+ \begin{bmatrix} 0 \\ 1 \\ 1\end{bmatrix} =A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} +
A \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} =
A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}
$$
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