I have a pretty basic question. Let's consider the linear transformation $\varphi = \begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}$ between the $2$-dimensional vector spaces $A$ and $B$ which are each isomorphic to $\mathbb{Z}\oplus \mathbb{Z}$, thus consider the map $$\mathbb{Z}\oplus \mathbb{Z} \to \mathbb{Z}\oplus \mathbb{Z}$$ and let's call it $\phi$ and let's assume we have a basis $\mathbb{Z}\oplus \mathbb{Z} =\langle a,b\rangle$.
The kernel $\ker \phi $ is given by $$ \ker \phi =\left\{ \lambda\begin{pmatrix}1 \\ -1 \end{pmatrix} \mid \lambda \in \mathbb{Z} \right\}$$ thus $\dim \ker \phi = 1$.
My question is: why can i express the kernel with respect to the generators $a,b$ by $$\ker \phi = \langle a-b\rangle?$$
I remember seeing this a few times but apparently there's something i am missing and i'd love to know why the last expression is valid. Clearly any element of the kernel is a linear combination of $a$ and $b$, but why is $\langle a-b\rangle$ generating $\ker \phi$?
Or rather, what's the relation between the solution $(1 -1)^\top$ of $\ker \phi$ and $a-b$ precisely?
Thanks for any help!