Orthogonal Group
For every dimension n>0, the orthogonal group O(n) is the group of n×n orthogonal matrices. These matrices form a group because they are closed under multiplication and taking inverses. Thinking of a matrix as given by n^2 coordinate functions, the set of matrices is identified with R^(n^2). The orthogonal matrices are the solutions to the n^2 equations A A^T = I, where I is the identity matrix, which are redundant. Only n(n + 1)/2 of these are independent, leaving n(n - 1)/2 "free variables." In fact, the orthogonal group is a smooth n(n - 1)/2-dimensional submanifold.
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