Macroeconomics Exam Review 20

Macroeconomics Exam Review 20 - Assume that the coalitions...

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Unformatted text preview: Assume that the coalitions are ordered so that 1 has the smallest number of players of any of the coalitions 1 , 2 ,... , . This implies that no coalition 2 , 3 ,... , is a subset of 1 and ( 1 ) = 0 for every = 2 , 3 ,... , (1.17) Using (1.39), 1 can be expressed as a linear combination of the other games, 1 = 1 / 1 =2 (1.18) Substituting (1.40) this implies that 1 ( 1 ) = 0 whereas ( ) = 1 for every by definition. This contradiction establishes that the set is linearly independent. 1.136 If is a subspace, then and x 1 = with = 0 and x 1 = (Exercise 1.122). Therefore is linearly dependent (Exercise 1.133). 1.137 Suppose x has two representations, that is x = 1 x 1 + 2 x 2 + ... + x x = 1 x 1 + 2 x 2 + ... + x Subtracting = (...
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