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Dimension of SO (n) and its generators - Mathematics Stack Exchange The generators of SO(n) S O (n) are pure imaginary antisymmetric n × n n × n matrices How can this fact be used to show that the dimension of SO(n) S O (n) is n(n−1) 2 n (n − 1) 2? I know that an antisymmetric matrix has n(n−1) 2 n (n − 1) 2 degrees of freedom, but I can't take this idea any further in the demonstration of the proof Thoughts?
Fundamental group of the special orthogonal group SO(n) Question: What is the fundamental group of the special orthogonal group SO(n) S O (n), n> 2 n> 2? Clarification: The answer usually given is: Z2 Z 2 But I would like to see a proof of that and an isomorphism π1(SO(n),En) → Z2 π 1 (S O (n), E n) → Z 2 that is as explicit as possible I require a neat criterion to check, if a path in SO(n) S O (n) is null-homotopic or not Idea 1: Maybe
In a family with two children, what are the chances, if one of the . . . For example, suppose there is a social science study on 2 child families with at least 1 daughter-- in this situation, about 1 3 of the families will be daughter-daughter, 1 3 will be daughter-son, and 1 3 will be son-daughter You have to consider the full probability space of two trials (d-d,d-s,s-d,s-s) and eliminate the s-s possibility
Boy Born on a Tuesday - is it just a language trick? The only way to get the 13 27 answer is to make the unjustified unreasonable assumption that Dave is boy-centric Tuesday-centric: if he has two sons born on Tue and Sun he will mention Tue; if he has a son daughter both born on Tue he will mention the son, etc
Improper integral of sin(x) x from zero to infinity I was having trouble with the following integral: ∫∞ 0 sin(x) x dx ∫ 0 ∞ sin (x) x d x My question is, how does one go about evaluating this, since its existence seems fairly intuitive, while its solution, at least to me, does not seem particularly obvious
Prove that the manifold $SO(n)$ is connected The question really is that simple: Prove that the manifold SO(n) ⊂ GL(n,R) is connected it is very easy to see that the elements of SO(n) are in one-to-one correspondence with the set of orthonormal basis of Rn (the set of rows of the matrix of an element of SO(n) is such a basis) My idea was to show that given any orthonormal basis (ai)n1 in Rn there's a continuous deformation from (ai
What does versus mean in the context of a graph? I would agree with the rule " [dependent] versus [independent] " The word "versus" can mean "compared with," and it more frequently makes sense to compare a dependent value with its associated independent value, because well, the independent variable doesn't really "care" about the existence of the dependent variable, but the converse relationship is by definition
Universal covering group and fundamental group of $SO(n)$ If H H is a topological group which is both path-connected and locally path-connected (i e a connected Lie group such as SO(n) S O (n)), then any path-connected cover of H H inherits a unique group structure making the covering map a group homomorphism In fact for any such cover p: G → H p: G → H,we have ker(p) ≅π1(H) p∗(π1(G)) k e r (p) ≅ π 1 (H) p ∗ (π 1 (G)) This