How to intuitively explain that if kernel(L) equal to the set containing the zero vector, then L is one-to-one? Whenever I try to learn about a relationship, I try to reason intuitively why a theorem or lemma should make sense. I know often times this is increasingly difficult to achieve. However, I have the following:
Let $ L: \mathbb{V} \to \mathbb{W} $ be a linear mapping. $L$ is one-to-one if and only if Ker($L$) = $\{\vec{0}\}$
I can intuitively explain the forward direction, that if $L$ is one-to-one, then only one value can possibly map to $\vec{0}$ and that must be $\vec0$. 
When I consider the other direction, it is hard for me to intuitively explain that the property of a linear mapping being one-to-one follows naturally from the fact that its kernel contains only $\vec0$. I know that the algebraic proof of this is trivial, but I feel like without a more intuitive understanding, I am misunderstanding something about linear mappings and hence not fully appreciating the importance of this relationship.
Thanks in advance.
EDIT:
The question is, can I explain the logic behind this principle without implicitly referring to the proof?

Intuitively, you can think of it as follows. Because the map is linear, the only way two values get mapped to the same thing is if they differ by an element of the kernel. This is because $A(x) = A(y)$ implies $A(x) - A(y) = A(x-y) = 0$ so that $x-y$ is in the kernel.
If the kernel only contains $0$, we're fine because the only way $x-y$ is in the kernel is if $x-y=0$, or that $x=y$, and the map is 1-1. But if the kernel contains other elements, say there exists $0 \neq k \in Ker(A)$, then for any $x$ in your space, $A(x + k) = A(x) + A(k) = A(x)$ so that $x+k$ and $x$ get mapped to the same value, which violates being 1-1.
Essentially, because the map is linear, you can "shift" elements around by elements of the kernel without affecting the result of the map. This will cause issues being 1-1 unless the only element you can "shift" by is $0$.
