Mapping of zero vector under a linear transformation Let $T$ be a linear transformation from a vector space $V$ to $W$ that are defined over a field $\mathbb F$. 
Now, $T(\mathbf0)=\mathbf0$. 
Which implies the zero vector in $V$ can't move or be transformed to a new vector in $W$. Why is it so? 
Does it also have a geometrical significance? 

$T(x+0_V)=T(x) + T(0_V)$ for linearity.
But $x+0_V = x$ because $0_V$ is the neutral element in $V$ wrt vector addition.
So it must be $T(x) = T(x) + T(0_V)$.
It follows that $T(0_V)$ must be the neutral element in $W$ wrt vector addition, that is $T(0_V) = 0_W$ 
