James wrote:
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?
Taken from:
http://en.wikipedia.org/wiki/Rotation_group
"By definition, a rotation about the origin is a linear transformation
that preserves the length of vectors, and also preserves the
orientation, or handedness, of space."
Yes, at this point the text is only about rotations in R^3, but below
you can read:
"The rotation group generalizes quite naturally to n-dimensional
Euclidean space, 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.
In R^3, that's an alternative way of defining "rotation".
So what does "rotation" mean in R^n?
I answered that question above.
Then, why does SO(n) give all rotations in R^n?
Be cause, *by definition* the rotations in R^n are the elements of
SO(n).
Best regards,
Jose Carlos Santos
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