Math 419H, Spring 2017 -- HW 1 Hint
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Q: We are wondering what bi-linear forms were, because they seem necessary to solve 1.4

A: A bilinear form is a type of dot-product on R^n defined by <v,w> = v^t Q w for a matrix Q.

The group O_1,1 is the set of 2x2 real matrices preserving the dot product of Q = diagonal matrix with diagonal entries 1,-1. That is, P with PQP^t = Q. The column vectors of P are an orthonormal basis of R^2 with respect to the Q-dot product, namely:
     <e_1,e_1> = 1 , <e_2,e_2> = -1 , <e_1,e_2> = <e_2,e_1> = 0 .
This dot product is used in special relativity, and O_1,1 is related to the Lorentz group.
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Q: The rotation by angle theta around a pole corresponds to a point on the longitude circle at cos(theta)/2 or 2(theta). Which one is correct? 

A: The rotation of angle theta in SO(3) corresponds to the SU_2 element:

     P = cos(theta/2) + sin(theta/2)p .

You can see this because P and -P give the same rotation (same conjugation action). Thus, the full range of rotations 0 < theta < 2 pi corresponds to only half the latitude circle.

Traveling by angle alpha around the longitude circle corresponds to a rotation of angle theta = 2 alpha in SO(3).