Math 6310 • Algebra

Let \(\mathcal{N}\) denote the set of two-by-two real matrices with determinant zero, let
\(\operatorname{SL}(2,\mathbb{R})\) denote the set of two-by-two matrices of determinant one, and
finally let \(\operatorname{GL}(2,\mathbb{R})\) denote the set of two-by-two matrices with
nonzero determinant.
\begin{enumerate}
\item[(a)] Show that both \(\operatorname{GL}(2,\mathbb{R})\) and \(\operatorname{SL}(2,\mathbb{R})\)
act on \(\mathcal{N}\) by conjugation.
\item[(b)] Describe the orbits of \(\operatorname{GL}(2,\mathbb{R})\) on \(\mathcal{N}\).
\item[(c)] Describe the orbits of \(\operatorname{SL}(2,\mathbb{R})\) on \(\mathcal{N}\).
\end{enumerate}