# Probs7 - Physics 316 Problem Set 7 due Tuesday November 15...

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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 co-ordinate x 1 of ξ is evidently a 1-form in ξ space, as are all the other co-ordinates. Exterior products of these 1-forms are the 2-forms x i ∧ x j . There are n 2 2-forms 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....
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Probs7 - Physics 316 Problem Set 7 due Tuesday November 15...

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