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Macroeconomics Exam Review 180

# Macroeconomics Exam Review 180 - Solutions for Foundations...

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3.238 Let a 𝑗 , 𝑗 = 1 , 2 , . . . , 𝑚 denote the rows of 𝐴 . Each a 𝑖 defines linear functional 𝑔 𝑗 ( 𝑥 ) = a 𝑗 𝑥 on 𝑛 , and c defines another linear functional 𝑓 ( 𝑥 ) = c 𝑇 x . Assume that 𝑓 ( 𝑥 ) = c 𝑇 x = 0 for every x 𝑆 where 𝑆 = { x : 𝑔 𝑗 ( x ) = a 𝑖 x = 0 , 𝑗 = 1 , 2 , . . ., 𝑚 } Then the system 𝐴𝑥 = 0 has no solution satisfying the constraint c 𝑇 x > 0. By Exercise 3.20, there exists scalars 𝑦 1 , 𝑦 2 , . . . , 𝑦 𝑚 such that 𝑓 ( x ) = 𝑚 𝑗 =1 𝑦 𝑗 𝑔 𝑗 ( x ) or c = 𝑚 𝑗 =1 𝑦 𝑗 𝑎 𝑗 = 𝐴 𝑇 y That is y = ( 𝑦 1 , 𝑦 2 , . . . , 𝑦 𝑚 ) solves the related nonhomogeneous system 𝐴 𝑇 y = c Conversely, assume that 𝐴 𝑇 y = c for some 𝑦 ∈ ℜ 𝑚 . Then c 𝑇 x = 𝑦𝐴𝑥 = 0 for all 𝑥 such that 𝐴𝑥 = 0 and therefore there is no solution satisfying the constraint c 𝑇 x = 1. 3.239 Let 𝑆 = { z : z = 𝐴 x , x ∈ ℜ } the image of 𝑆 . 𝑆 is a subspace. Assume that system I has no solution, that is
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