Esercizio 15   Provare che i seguenti insiemi sono sottogruppi di $GL_{2}(\mathbb{R} )$:
a)
$H_{1}=\{\left(\begin{array}{cc}a&b\\
-b&a\end{array}\right)\vert\,a,b\in \mathbb{R} ,\,a^{2}+b^{2}\neq 0\};$
b)
$H_{2}=\{\left(\begin{array}{cc}a&b\\
c&d\end{array}\right)\vert\,a,b,c,d\in \mathbb{R} ,\,ad-bc\in \mathbb{Q} ^{*}\};$
c)
$H_{3}=\{\left(\begin{array}{cc}a&b\\
0&1\end{array}\right)\vert\,a,b\in \mathbb{R} ,\,a\neq 0\};$
d)
$H_{4}=\{\left(\begin{array}{cc}a&b\\
0&a^{-1}\end{array}\right)\vert\,a,b\in \mathbb{R} ,\,a\neq 0\}.$


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