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Macroeconomics Exam Review 23

Macroeconomics Exam Review 23 - Solutions for Foundations...

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1.146 We have previously shown that the set 𝑈 is linearly independent (Exercise 1.135). the space 𝒢 𝑁 has dimension 2 𝑛 1 (Exercise 1.141). There are 2 𝑛 1 distinct T-unanimity games 𝑢 𝑇 in 𝑈 . Hence 𝑈 spans the 2 𝑛 1 space 𝒢 𝑁 . Alternatively, note that any game 𝑤 ∈ 𝒢 𝑁 can be written as a linear combination of T-unanimity games (Exercise 1.75). 1.147 Let 𝐵 = { x 1 , x 2 , . . . , x 𝑚 } be a basis for 𝑆 . Since 𝐵 is linearly independent, 𝑚 𝑛 (Exercise 1.143). There are two possibilities. Case 1: 𝑚 = 𝑛 . 𝐵 is a set of 𝑛 linearly independent elements in an 𝑛 -dimensional space 𝑋 . Hence 𝐵 is a basis for 𝑋 and 𝑆 = lin 𝐵 = 𝑋 . Case 2: 𝑚 < 𝑛 . Since 𝐵 is linearly independent but cannot be a basis for the 𝑛 - dimensional space 𝑋 , we must have 𝑆 = lin 𝐵 𝑋 . Therefore, we conclude that if 𝑆 𝑋 is a proper subspace, it has a lower dimension than 𝑋 .
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