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Macroeconomics Exam Review 21

Macroeconomics Exam Review 21 - 1.140 Let = a 1 a 2 a and =...

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Unformatted text preview: 1.140 Let = { a 1 , a 2 ,... , a } and = { b 1 , b 2 ,... , b } be two bases for a linear space . Let 1 = { 1 } ∪ = { b 1 , a 1 , a 2 ,..., a } is linearly dependent (since 1 ∈ lin ) and spans . Therefore, there exists 1 , 2 ,... , and 1 such that 1 b 1 + 1 a 1 + 2 a 2 + ... + a = At least one ∕ = 0. Deleting the corresponding element a , we obtain another set ′ 1 of elements ′ 1 = { b 1 , a 1 , a 2 ,... , a − 1 , a +1 ,..., a } which is also spans . Adding the second element from , we obtain the +1 element set 2 = { b 1 , b 2 , a 1 , a 2 ,... , a − 1 , a +1 ,... , a } which again is linearly dependent and spans . Continuing in this way, we can replace vectors in with the vectors from while maintaining a spanning set. This process cannot eliminate all the vectors inmaintaining a spanning set....
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