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Macroeconomics Exam Review 25

# Macroeconomics Exam Review 25 - Solutions for Foundations...

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Then 𝐻 = x 0 + 𝑉 is an aﬃne set (Exercise 1.150) which strictly contains 𝐻 . This contradicts the definition of 𝐻 as a maximal proper aﬃne set. 5. Let x 1 / 𝑉 . By the previous part, x lin { x 1 , 𝑉 } . That is, there exists 𝛼 ∈ ℜ such that x = 𝛼 x 1 + v for some v 𝑉 To see that 𝛼 is unique, suppose that there exists 𝛽 ∈ ℜ such that x = 𝛽 x 1 + v for some v 𝑉 Subtracting 0 = ( 𝛼 𝛽 ) x 1 + ( v v ) which implies that 𝛼 = 𝛽 since x 1 / 𝑉 . 1.154 Assume x , y 𝑋 . That is, x , y ∈ ℜ 𝑛 and 𝑖 𝑁 𝑥 𝑖 = 𝑖 𝑁 𝑦 𝑖 = 𝑤 ( 𝑁 ) For any 𝛼 ∈ ℜ , 𝛼 x + (1 𝛼 ) y ∈ ℜ 𝑛 and 𝑖 𝑁 𝛼𝑥 𝑖 + (1 𝛼 ) 𝑦 𝑖 = 𝛼 𝑖 𝑁 𝑥 𝑖 + (1 𝛼 ) 𝑖 𝑁 𝑦 𝑖 = 𝛼𝑤 ( 𝑁 ) + (1 𝛼 ) 𝑤 ( 𝑁 ) = 𝑤 ( 𝑁 ) Hence 𝑋 is an aﬃne subset of 𝑛 . 1.155 See Exercise 1.129. 1.156 No. A straight line through any two points in 𝑛 + extends outside 𝑛 + . Put differently, the aﬃne hull of 𝑛
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