functions – Is that a linear transformation?

I came across the question if
f(x) = x + y (x and y are vectors)
is a linear transformation.

I would answer that this is not a linear transformation because if y would be anything else than the zero vector then for x = zero vector, the resulting vector would not be the zero vector and thus, the origin would not stay in place.

But if y would be the zero vector, would this count as a linear transformation?