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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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    @JulianKuelshammer: Okay, I put it as an answer. I've never used chat rooms before, so maybe you can post there instead.2013-06-14

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