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Unformatted text preview: TA: Yankun Wang Econ 617 Problem Set 2 Solution Key 1. [Linear Dependence and Independence] (a) Let S = { e 1 , e 2 , e 3 } be the set of unit vectors in R 3 . Let T = { x 1 , x 2 , x 3 } be a set of vectors in R 3 , de f ned by: x 1 = e 1 + e 2 , x 2 = e 2 + e 3 , x 3 = e 3 + e 1 Is T a set of linearly independent vectors in R 3 ? Explain. Solution: Claim: T is a set of linearly independent vectors in R 3 . Proof: Suppose there exists λ 1 , λ 2 and λ 3 such that: λ 1 x 1 + λ 2 x 2 + λ 3 x 3 = 0 , i.e. : λ 1 ⎛ ⎝ 1 1 ⎞ ⎠ + λ 2 ⎛ ⎝ 1 1 ⎞ ⎠ + λ 3 ⎛ ⎝ 1 1 ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ . ⇒ ⎛ ⎝ λ 1 + λ 3 λ 1 + λ 2 λ 2 + λ 3 ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ ⇒ λ 1 = 0; λ 2 = 0; λ 3 = 0 . Thus by de f nition, T is a set of linearly independent vectors in R 3 . (b) Let S = { e 1 , e 2 , e 3 } be the set of unit vectors in R 3 . Let T = { x 1 , x 2 , x 3 } be a set of vectors in R 3 , de f ned by: x 1 = e 1 , x 2 = e 1 + 2 e 2 , x 3 = e 1 + 2 e 2 + 3 e 3 Is T a set of linearly independent vectors in R 3 ? Explain....
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This note was uploaded on 08/29/2009 for the course ECON 617 taught by Professor Staff during the Fall '08 term at Cornell.
 Fall '08
 STAFF

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