A linear transformation $T: \Bbb Z_p \times \Bbb Z_p \to \Bbb Z_p \times \Bbb Z_p $, where $p$ is a prime, is defined by $T(v)=s^{-1}vs$. The minimal polynomial is $m(x)=x^2+x+1$. Also $u=T(t)=s^{-1}ts$. Then $\{t,u\}$ is a basis for $\Bbb Z_p \times \Bbb Z_p$. Then can we say $T(t^i u^j)= t^{-j} u^{i-j}$?
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