This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Physics 316, Problem Set 7, due Tuesday November 15 in class 1. linear independence of basic exterior forms cf. p. 167 problem 6. If a vector space has a basis e 1 ... e n then any vector ξ can be represented as ξ = x 1 e 1 + x 2 e e + ... + x n e n . The coordinate x 1 of ξ is evidently a 1form in ξ space, as are all the other coordinates. Exterior products of these 1forms are the 2forms x i ∧ x j . There are n 2 2forms of this type, but only the n ( n 1) / 2 for which j < i are really distinct (since x j ∧ x i = x i ∧ x j ). The question arises whether these n ( n 1) / 2 forms are really distinct themselves. a) Which of the x 1 ∧ x 2 ( e i , e j ) are nonzero for the various i < j ? b) Are the x i ∧ x j with i < j all linearly independent? Why or why not? Solution: a) Each e i has x i = 1 and x j = 0 for j 6 = i . So, the e ’s that give a nonzero x 1 ∧ x 2 are e 1 and e 2 . b) If x i ∧ x j for i < j are linearly dependent, we can form a linear combination of them that vanishes for all ξ 1 , ξ 2 : That is for some a ij , and all ξ 1 , ξ 2 . 0 = ω 2 ( ξ 1 , ξ 2 ) ≡ X i<j a ij x i ∧ x j ( ξ 1 , ξ 2 ) We can clearly find some ξ ’s for which this sum would vanish. The question is whether we can make it’s for which this sum would vanish....
View
Full Document
 Spring '10
 Pucker
 Physics, Linear Algebra, mechanics, Vector Space, basis, ξ

Click to edit the document details