- #1
SNOOTCHIEBOOCHEE
- 145
- 0
Homework Statement
Prove algebraically that a real 2x2 matrix [tex]\left(\begin{array}{cc}a&b\\c&d\end{array}\right)[/tex] represents a rotaion iff it is in SO2
Homework Equations
In case you are used to different notation, SO2= {A[tex]\in[/tex] GLn(R)| AtA=I, Det A=1}
The Attempt at a Solution
ok since this is an iff statement, we have to show both directions.
Case1: if the matrix is in SO2 then it represents a rotation
so we know that [tex]\left(\begin{array}{cc}a&b\\c&d\end{array}\right)[/tex] * [tex]\left(\begin{array}{cc}a&c\\b&d\end{array}\right)[/tex]= [tex]\left(\begin{array}{cc}1&0\\0&1\end{array}\right)[/tex]
also ad-bc=1
also if its helpful [tex]\left(\begin{array}{cc}a&b\\c&d\end{array}\right)[/tex] * [tex]\left(\begin{array}{cc}a&c\\b&d\end{array}\right)[/tex]= [tex]\left(\begin{array}{cc}a2+b2&ac+bd\\ca+db&b2+d2\end{array}\right)[/tex]
I know i can set this equal to the identity and probably solve for some stuff. but how exactly do i prove that it is a roation? moreover i am completley lost in the other direction.
Edit: maybe i figured it out??!?
Other direction: If [tex]\left(\begin{array}{cc}a&b\\c&d\end{array}\right)[/tex] is a rotation, then it is in SO2
every rotaion through an angle theta can be written as cos -sin sin cos. just show that its transpose * it =1? and that its det 1? which is basically trivial... so I am think i did this second part wrong.