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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?
Finding the ages of father and son without using algebraic equations Therefore, the present ages of the father and son are 45 − 5 = 40 45 − 5 = 40, and 15 − 5 = 10 15 − 5 = 10 years, respectively The part of the solution that I cannot understand is how is the statement - But by the question, this sum is 4 times the age of the son - arrived at? Any help will be much appreciated
problem solving - Diophantus Lifespan - Mathematics Stack Exchange "The son lived exactly half as long as his father" is I think unambiguous Almost nothing is known about Diophantus' life, and there is scholarly dispute about the approximate period in which he lived There is no reason to think that the problem has a historical basis
lie groups - Lie Algebra of SO (n) - Mathematics Stack Exchange Welcome to the language barrier between physicists and mathematicians Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators So for instance, while for mathematicians, the Lie algebra so(n) consists of skew-adjoint matrices (with respect to the Euclidean inner product on Rn), physicists prefer to multiply them by I think − i (or maybe
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
Is $SO(n)$ actuallly the same as $O(n)$? - Mathematics Stack Exchange In the current problem, X X is a member of so(n) s o (n) (or of o(n) o (n)) Now since all members of o(n) o (n) are anti-symmetric matrices which have all its diagonal elements zero, their traces are all zero: Tr X = 0 Tr X = 0 This implies that det eX = 1 det e X = 1, so all members of O(n) O (n) have determinant 1 1, none of them have determinant −1 − 1 This further means that SO(n
How to find the difference between the sons and mothers age if it . . . A son had recently visited his mom and found out that the two digits that form his age (eg :24) when reversed form his mother's age (eg: 42) Later he goes back to his place and finds out that this whole 'age' reversed process occurs 6 times And if they (mom + son) were lucky it would happen again in future for two more times
Fundamental group of the special orthogonal group SO(n) Question: What is the fundamental group of the special orthogonal group , n> 2? Clarification: The answer usually given is: Z2 But I would like to see a proof of that and an isomorphism π1(SO(n), En) → Z2 that is as explicit as possible I require a neat criterion to check, if a path in SO(n) is null-homotopic or not Idea 1: Maybe it is helpful to think of SO(n) as embedded in SO(n + 1
general topology - proving that $SO (n)$ is path connected . . . There are some details here that needs ironing out, but this approach should work with the results you already have: Take any two points in SO(n) S O (n), map them via quotient map q q to SO(n) SO(n − 1) S O (n) S O (n − 1) Connect them via a path in that space If the path doesn't go through the point q(SO(n − 1)) q (S O (n − 1)), lift the path via q−1 q − 1 Rejoice However
What is the relationship between SL (n) and SO (n)? To add some intuition to this, for vectors in Rn, SL(n) is the space of all the transformations with determinant 1, or in other words, all transformations that keep the volume constant This is because the determinant is what one multiplies within the integral to get the volume in the transformed space SO(n) is the subset in which the transformation is orthogonal (RTR = I) Such matrices