Let $U$ be a finite dimensional vector space over $\mathbb{C}$ and let
$T\_1 : U → U,\dots , T\_n : U → U$ be pairwise commuting linear maps.

I was to show that there exists a $v ∈ V$ that is an eigenvector of all $T\_i$
The hint given to me was to first prove that the eigenspaces of $T\_i$ are preserved by $T\_j$, but I don't quite get it

Could anyone help please?