Show that the Image of $0$ under a Linear Mapping is also $0$

Question

In my book of Linear Algebra, I have the following exercises:

Let $T: V \to W$ be a linear map from one vector space to another. Show that $T(0) = 0$.

I'm somehow having a block.

For me, it is natural to express it like: If $T(0)$ were not in $Ker(T)$ and thus contradiction, because $Ker(T)$ is a subspace by default (the way it is constructed).

How can we show that with more elementarily? (with less abstraction.)

Answer 1

Hint: Begin with $0=0+0\implies T(0)=T(0+0)$.