The null space of a linear map $\mathcal A:U\to V$ is:
$$ \text{Null}(\mathcal A)=\{u\in U|\mathcal A(u)=0_V\} $$
For $\mathcal A$ to be linear, we require that $\text{Null}(\mathcal A)=\{0_U\}$, i.e. the addition identity element on linear space $U$.
Question: why does $\text{Null}(\mathcal A)=\{0_U\}$ mean that the null space is dimension 0? In other words, the nullity of the matrix representation is 0. For me, as there is one element in the null space ($0_U$), shouldn't the dimension be 1?