Do all linear transformations map the zero vector to the zero vector?

Question

For example, a linear transformation T that maps between two vector spaces V and W. Does T map the zero vector of V to the zero vector of W? Is that a rule for linear transformations? Or no? I've never seen a rule like this explicitly stated, but at the same time I can't think of an example where this wouldn't be a case.

Answer 1

It's easy to prove that a linear transformation $T: V\to W$ maps $0_V$ to $0_W$. It follows from the definition easily:

$$T(\vec{0}_V)=T(\vec{0}_V+\vec{0}_V)=T(\vec{0}_V)+T(\vec{0}_V)$$ $$T(\vec{0}_V)=\vec{0}_W$$

Here are the properties of a vector space that we're using: the zero vector is the additive identity which means that if we add it to any other vector, it gives it back. Then we're using the fact that a linear transformation preserves addition. And at the end, we're adding $-T(\vec{0}_V)$ to both sides and after cancellation we're left with the last equation which is what we wanted to prove.