- Book recommendations for linear algebra - Mathematics Stack Exchange
I have been wanting to learn about linear algebra (specifically about vector spaces) for a long time, but I am not sure what book to buy, any suggestions?
- lie groups - Lie Algebra of SO (n) - Mathematics Stack Exchange
Where a, b, c, d ∈ 1, …, n a, b, c, d ∈ 1,, n And so(n) s o (n) is the Lie algebra of SO (n) I'm unsure if it suffices to show that the generators of the
- 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
- Homotopy groups O(N) and SO(N): $\\pi_m(O(N))$ v. s. $\\pi_m(SO(N))$
I have known the data of $\\pi_m(SO(N))$ from this Table: $$\\overset{\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\textbf{Homotopy groups of
- 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?
- 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
- 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
- Prove that the manifold $SO(n)$ is connected
The question really is that simple: Prove that the manifold SO(n) ⊂ GL(n,R) S O (n) ⊂ G L (n, R) is connected it is very easy to see that the elements of SO(n) S O (n) are in one-to-one correspondence with the set of orthonormal basis of Rn R n (the set of rows of the matrix of an element of SO(n) S O (n) is such a basis) My idea was to show that given any orthonormal basis (ai)n1 (a i
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