Where ReLU Breaks Additivity

A single layer computes $f(x) = \mathrm{ReLU}(Wx)$, where $\mathrm{ReLU}$ is applied entrywise and

$W = \begin{bmatrix} 1 & -2 \\ 3 & 1 \end{bmatrix}$

Take the two inputs

$u = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \qquad v = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$

If $f$ were a linear map, then $f(u+v)$ and $f(u) + f(v)$ would be identical and their difference would be the zero vector. Compute the difference and see what actually happens:

$f(u+v) - \big(f(u) + f(v)\big)$