Let $T$ be linear transformation from $V$ to $W$. I know how to prove the result that nullity$(T) = 0$ if and only if $T$ is an injective linear transformation. But I still don't intuitively understand why the kernel only containing the zero vector means that $T$ is injective, and vice versa. In contrast, the relation between the image of $T$ and condition of being surjective is easy to see, since in order to map to all of the elements of $W$ the image of $T$ must have the same dimension as $W$. This can intuitively be seen with a diagram of the mapping from $V$ to $W$, for example. I can't really imagine a diagram that plainly shows the injective condition.
In short, what about nullity$(T) = 0$ imples that $T$ is a one-to-one function?