SO(n) and O(n) questions

Dear all,

I am trying to understand more the groups SO(n) and O(n) geometrically. I think I have a good understanding of them algebraically.

Well, my first question is : What does the word "rotation" mean in R^n? In R^3, the map A : R^3 ---> R^3 is a rotation if it preserves distances between points, fixes the origin, fixes some non-zero vector, and represents a 2-dimensional rotation in the plane perpendicular to this fixed vector.

So what does "rotation" mean in R^n? Then, why does SO(n) give all rotations in R^n? I know why SO(3) gives all rotations in R^3 (proof in Artin's algebra page 129). Also, geometrically, what is O(n)? I would like to learn even more about SO(n) and O(n) geometrically, but I am not sure which questions to ask. Can you recommend a good reference for me to learn?


Sincerely,

James
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