Question

Question: (c) Show that everyco-set in$\mathit{T}\mathbf{/}\mathit{l}$ can be written in the form$\left(\begin{array}{cc}a& 0\\ 0& a\end{array}\right)\mathbf{+}\mathit{I}$

Short answer

$T/l=\left\{\left[\begin{array}{cc}a& 0\\ 0& c\end{array}\right]+l|a\in \mathrm{ℝ}\right\}$

Step-by-step solution

Obtain co-sets

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

$T/l=\left\{\left[\begin{array}{cc}a& 0\\ 0& a\end{array}\right]+I|a\in \mathrm{ℝ}\right\}$

These are the co-sets T/lin in the form of $\left(\begin{array}{cc}a& 0\\ 0& a\end{array}\right)+I.$