In our notes the following is written:
Let $M \subseteq \mathbb R^{n}$ and define $\operatorname{dim} (M):=\operatorname{dim}(\operatorname{aff} (M))$
it then goes on to say that $\operatorname{aff} (M)=a+M$ with $a \in M$, it follows that $\operatorname{dim}(M)=\operatorname{max} \{d \in \mathbb N_{0}: \exists a, x_{1},...,x_{d} \in M \operatorname{with} \{ x_{i} -a \}_{\{1\leq i \leq d\}} \operatorname{linearly independent}\}$
I am uncertain whether it is true that $\operatorname{aff} (M)=a+M$ WITH $a \in M$. I am having a hard time to understand why $a \in M$, as well as why the dimension of the affine hull is given by $\operatorname{dim}(M)$