John Stillwell - Naive Lie Theory.pdf - Index of

3.2 The orthogonal and special orthogonal groups 53
Thus a final position A of R n can be realized by a “rotation” in the
everyday sense of the word only if SO(n) is path-connected.
Path-connectedness of SO(n). For any n, SO(n) is path-connected.
Proof. For n = 2 we have the circle SO(2), which is obviously pathconnected
(Figure 3.1). Now suppose that SO(n − 1) is path-connected
and that A ∈ SO(n). It suffices to find a path in SO(n) from 1 to A, because
if there are paths from 1 to A and B then there is a path from A to B.
cos θ + isinθ
θ
1
Figure 3.1: Path-connectedness of SO(2).
This amounts to finding a continuous motion taking the basis vectors
e 1 ,e 2 ,...,e n to their final positions Ae 1 ,Ae 2 ,...,Ae n (the columns of A).
The vectors e 1 and Ae 1 (if distinct) define a plane P, so, by the pathconnectedness
of SO(2), we can move e 1 continuously to the position Ae 1
by a rotation R of P. It then suffices to continuously move Re 2 ,...,Re n to
Ae 2 ,...,Ae n , respectively, keeping Ae 1 fixed. Notice that
• Re 2 ,...,Re n are all orthogonal to Re 1 = Ae 1 , because e 2 ,...,e n are
all orthogonal to e 1 and R preserves angles.
• Ae 2 ,...,Ae n are all orthogonal to Ae 1 , because e 2 ,...,e n are all orthogonal
to e 1 and A preserves angles.
Thus the required motion can take place in the R n−1 of vectors orthogonal
to Ae 1 , where it exists by the assumption that SO(n−1) is path-connected.
Performing the two motions in succession—taking e 1 to Ae 1 and then
Re 2 ,...,Re n to Ae 2 ,...,Ae n —gives a path from 1 to A in SO(n). □
The idea of path-connectedness will be explored further in Sections 3.8
and 8.6. In the meantime, the idea of continuous path is used informally in