Questions about the irreducible representations Let $G$ be a finitely generated group, consider the irreducible representations $\{\rho:G\to GL(n,\mathbb{C})\}$ 
Fix $n^2$ elements $g_1,...,g_{n^2}\in G$ and assume there exists some irreducible representation $\rho$, s.t. $\rho(g_1),...,\rho(g_{n^2})$ span $M(n,\mathbb{C})$, I wonder if it's possible to exist some other irreducible representation $\rho'$, s.t. $\rho'(g_1),...,\rho'(g_{n^2})$ span $M(n,\mathbb{C})$?
And if $m_1,...,m_{n^2}$ span $M(n,\mathbb{C})$, if there always exists some irreducible representation $\rho$, s.t. $\rho(g_i)=m_i$? 
 A: Q1. You could, for example, define each $\rho'(g_i)$ to be $X^{-1}\rho(g_i)X$ for some fixed invertible matrix.
Q2. No this is not always possible. Remember that $\rho$ is a group homomorphism so  you cannot expect $g_1$ to be able to map to any element of a spanning set.
A: Let $E_{ij}$ denote the $n\times n$ matrix with 1 at $(i,j)$th position and  zero everywhere else. Clearly these $n^2$ matrices span  the vector space of all $n\times n$ matrices. Unfortunately all these matrices are singular (in fact rank 1)
Now any representation of any group should map the elements to non-singular matrices. So your question in general has NO as the answer.
To avoid this obvious troubling issue, let us take  all the $n^2$ matrices to be non-singular. Now we can always construct a basis by extending a linear independent set. So given any non-singular matrix there exists a basis that includes it.
Choose this non-singular matrix to have  $\pi$ as one of the entries and all other entries integers. (Or any matrix $A$ such that $A^m\neq I$ for any $m$).
Take $G$ now to be a finite group. For any element $g\in G$, $\rho (g)$ must be of finite order. And so $\rho(g)\neq A$. This proves that answer to your second question is Negative again (even if we allow reducuble representations).
