# Matrix representation of vector-valued function

Consider the following multivariate vector-valued function:

$f: \{0,1\}^3 \rightarrow \{0,1\}^3$

$f(\mathbf{x}) = ( f_1(\mathbf{x}), f_2(\mathbf{x}), f_3(\mathbf{x}) )$

with $\mathbf{x}=(x_1, x_2, x_3) \in \{0,1\}^3$ and $f_1, f_2, f_3: \{0,1\}^3 \rightarrow \{0,1\}^3$.

For example, we can define $f$ as: \begin{cases} f_1(\mathbf{x}) = x_2 + x_3 \\[1ex] f_2(\mathbf{x}) = x_3 \\[1ex] f_3(\mathbf{x}) = (x_1 + x_3) (2-x_1 - x_3) \end{cases} with $f(1,0,0)=(0,1,0)$

Is it possible to represent $f$ as a matrix? Something like:

$$A \times \left[ \begin{array}{cc} 1\\ 0\\ 0\\ \end{array} \right] = \left[ \begin{array}{cc} 0\\ 1\\ 0\\ \end{array} \right]$$

Thanks.

I don't think so. You don't have a linear function in $f_3(x)$ in your example.