# Isomorphic representations of $\mathbb Z$

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

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It means that you can find bases $\beta_1$ and $\beta_2$ of $V$ such that $[\rho(1)]_{\beta_1}$ (the coordinate matrix of $\rho(1)$ relative to $\beta_1$) and $[\rho'(1)]_{\beta_2}$ (the coordinate matrix of $\rho'(1)$ relative to $\beta_2$) are identical. It doesn't literally mean "the same linear map", it means "the same linear map up to automorphisms of $V$". –  Arturo Magidin Jan 24 '12 at 22:24
It literally means the same matrix with the same entries, but possibly using different bases for $\rho(1)$ and $\rho'(1)$. –  Grumpy Parsnip Jan 24 '12 at 22:25
@GrumpyParsnip Please consider converting your comment into an answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. –  Julian Kuelshammer Jun 13 '13 at 21:16
@JulianKuelshammer: Okay, I put it as an answer. I've never used chat rooms before, so maybe you can post there instead. –  Grumpy Parsnip Jun 14 '13 at 2:08

It literally means the same matrix with the same entries, but possibly using different bases for $\rho(1)$ and $\rho′(1).$