
#1
Jun2411, 08:18 AM

P: 205

I have this problem(from Tensor Analysis on Manyfolds by Bishop and Goldberg): prove that
[itex]e_1^ e_2 + e_3^e_4[/itex] is not decomposable when the dimension of the vector space is greater than 3 and e_i are basis vectors. I solved it by mounting a set of 6 equations with 8 unknows and studying the different posibilities cheking that each one is not solvable. Is there any nicer way to tackle this problem? if so please let me know 



#2
Jun2411, 08:56 AM

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hi facenian!
(use "\wedge" in latex ) so express a and b in terms of the basis 



#3
Jun2411, 09:02 AM

P: 205





#4
Jun2411, 09:50 AM

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wedge product
how about [itex]a\wedge (e_1\wedge e_2 + e_3\wedge e_4)[/itex] ?




#5
Jun2811, 10:34 AM

P: 205





#6
Jun2811, 10:43 AM

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hi facenian!
no, i'm using the same a as before (in a∧b, which you're trying to prove it isn't) so let a = ∑_{i} x_{i}e_{i} 



#7
Jun2811, 10:56 AM

P: 205

I'm sorry I did not explained it correctly I should have said:
you mean, let [itex]a=\sum_i x_{i} e_i[/itex] and then conclude tha [itex]a[/itex] must be null because we are left with a linear conbination of basic vectors of the form [itex] \sum x_i e_i\wedge e_j\wedge e_k=0[/itex] .Please let me know if that's what you meant and/or if I'm correct 



#8
Jun2811, 11:00 AM

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yes 



#9
Jun2811, 11:03 AM

P: 205

thank you very much tinytim your method is much better than mine!



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