Question

Question: ( c) Show that there are infinitely many distinct co-sets in$\mathit{S}\mathbf{/}\mathit{I}$, one for each pair in$\mathit{ℝ}\mathbf{\times }\mathit{ℝ}$

Short answer

$S/I=\left\{\left[\begin{array}{cc}a& 0\\ 0& c\end{array}\right]+l|\left(a,c\right)\in \mathrm{ℝ}\times \mathrm{ℝ}\right\}$

Step-by-step solution

Obtain co-sets

Consider for any , $\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]\in S$, so that

$\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]=\left[\begin{array}{cc}a& 0\\ 0& c\end{array}\right]+\left[\begin{array}{cc}0& b\\ 0& 0\end{array}\right]\in \left[\begin{array}{cc}a& 0\\ 0& a\end{array}\right]+l$

Thus,

$S/I=\left\{\left[\begin{array}{cc}a& 0\\ 0& c\end{array}\right]+l\left|\right(a,c)\in \mathrm{ℝ}\times \mathrm{ℝ}\right\}$

This is the co-sets in in S/I in$\mathrm{ℝ}\times \mathrm{ℝ}$