the point is, that is a linear map (in the sense of vector spaces), and linear maps always preserve the identity element of the vector space, as an abelian group (linear maps are abelian group homomorphisms that preserve scalar multiplication, as well).

it seems that the linear algebra might be throwing you off, a bit. it is accepted as so basic a fact of linear algebra (that the 0-vector of a vector space is in the kernel of a linear map), that most authors just state it without proof.