I've read a statement in my notes that I am confused about:
Representations $\rho, \rho' : \mathbb Z \to \mathrm{GL}(V)$ are isomorphic iff we may choose bases such that $\rho(1)$ and $\rho'(1)$ are the same matrix.
I understand the relevance of $\rho(1)$ here, since specifying the image of $1$ determines the entire representation. I'm confused specifically about the meaning of "the same matrix". Does this mean "the same linear map", or more literally matrices $A$ and $B$ with $A_{ij} = B_{ij}$ for all $i,j$?
Thanks