can  rotate  rotate, reflect  rotate, reflect, scale 

get  angle, length, orient'n  angle, length  angle 
name  \( SO(\mathbb{R}^2) \)  \( O(\mathbb{R}^2) \)  \( CO(\mathbb{R}^2) \) 
can  rotate  rotate, reflect  rotate, reflect, scale 

get  angle, length, orient'n  angle, length  angle 
name  \( SO(\mathbb{R}^2) \)  \( O(\mathbb{R}^2) \)  \( CO(\mathbb{R}^2) \) 
Actions  Name  Formal Definition 

Rotations  \(SO(\mathbb{R}^n)\)  \( =\{ A : A^TA=I, \det(A)=1\} \subset GL \) 
Rotations & Reflections  \(O(\mathbb{R}^n)\)  \( =\{ A : A^TA =I, \det(A)=\pm 1\} \subset GL\) 
Rotations & Reflections & Scalings  \(CO(\mathbb{R}^n)\)  \( =\{ rA : A \in O(n), r \in \mathbb{R} \} \subset GL\) 
Volumepreserving  \(SL(\mathbb{R}^n)\)  \( =\{ A : \det(A)=1\} \subset GL\) 
All (linear)  \(GL(\mathbb{R}^n)\)  \( =\{ A : \det(A)\neq 0\}\) 
\(\cdots\)  G  \( =\{ A : \cdots~\text{some condition}~\cdots \} \subset GL\) 
\( (7.5,6.5) \in K\)  \( \{ (\mathrm{d}x,\mathrm{d}y), \mathrm{d}y = 1 \mathrm{d}x\} \) 
\( p \in K\)  \( \Sigma_p \subset T_p K\) 
\( \{(x,f(x))\} \subset K\)  \( \{\mathrm{d}y=f'(x)\mathrm{d}x\} \subset T_p K\) 
Compute \(\Delta = B^2  4AC\):

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