3
$\begingroup$

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

  • 6
    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 $V$".2012-01-24
  • 5
    It literally means the same matrix with the same entries, but possibly using different bases for $\rho(1)$ and $\rho'(1)$.2012-01-24
  • 0
    @GrumpyParsnip Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-13
  • 1
    @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

1 Answers 1