Macroeconomics Exam Review 116

Macroeconomics Exam Review 116 - 3.49 Let be a hyperplane...

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Unformatted text preview: 3.49 Let be a hyperplane in . Then there exists a unique subspace such that = x + for some x ∈ (Exercise 1.153). There are two cases to consider. Case 1: x / ∈ . For every x ∈ , there exists unique x ∈ ℜ such x = x x + for some ∈ Define ( x ) = x . Then : → ℜ . It is straightforward to show that is linear. Since = x + , x = 1 if and only if x ∈ . Therefore = { x ∈ : ( x ) = 1 } Case 2: x ∈ . In this case, choose some x 1 / ∈ . Again, for every x ∈ , there exists a unique x ∈ ℜ such x = x x 1 + for some ∈ and ( x ) = x is a linear functional on . Furthermore x ∈ implies = (Exercise 1.153) and therefore ( x ) = 0 if and only if x ∈ . Therefore = { x ∈ : ( x ) = 0 } Conversely, let be a nonzero linear functional in ′ . Let...
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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