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Final

MAT 150A Lecture Notes - Lecture 12: Unit Circle, Dihedral GroupExam


Department
Mathematics
Course Code
MAT 150A
Professor
José Simental Rodríguez
Study Guide
Final

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MAT 150A Karishma Carter Johnson October 24, 2018
Lemma
Let MMatn×n(R) then MOniff it’s columns form an orthonormal collection.
Proof. ) Assume Ais orthogonal
A= [a1|...|an]
At=
a1
a2
...
an
Since A is orthogonal, AtA=I
(AtA)ij =ai·aj=δij
=ai·aj=δij
⇒ {ai}form an orthonormal collection of vectors in Rn
today studying O2and SO2
Recall
Orthogonal Group
On={MMat(n×n)|M1=Mt}
SO(n) = O(n)SL(n)
A=a11 a12
a21 a22O2=a11
a21and a12
a22
are orthonormal
R2unit circle.
choose an arbitrary point on the unit circle θdegrees from the right
a11
a21=cos θ
sin θ
A=cos θsin θ
sin θcos θ
det(A) = 1 ASO2
ˆrefers to the rotation for initial point to 90 degrees counterclockwise
sin θ
cos θ
the point 90 degrees clockwise
1
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