SU(2) to SO(3)?

Question

If you've studied topology, you may know that SU(2) is the simply connected cover of SO(3). Here's a fun question: What is the map in terms of matrix coefficients? That is, if I give you the coefficients of an SU(2) matrix, produce the coefficients of the corresponding SO(3) matrix.

I don't recommend trying to solve this problem by analyzing determinantal expressions. Here's a hint if you're interested: Remember that we can identify SU(2) with the unit quaternions, and recall how we use this identification to construct a map SU(2)xSU(2) --->SO(4).

Answer 1

This problem is in Introduction to Smooth Manifolds by John M. Lee page 196 EX 8.35.

Basically you use determinates to prove this. So you need some linear algebra.