Homework 4 - Aaron Goldsmith Homework 3 Section 4.2 Algebra...

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Aaron Goldsmith Homework 3 Section 4.2 Algebra II 1 a) If { x 1 ,...,x n } is linearly dependent, then there exist scalars, not all zero, such that a 1 x 1 + ... + a n x n = 0 Let k be the largest index for which a k 6 = 0. Then, x k = - a - 1 k ( a 1 x 1 + ... + a k - 1 x k - 1 ) = - a - 1 k a 1 x 1 - ... - a - 1 k a k - 1 x k - 1 If x k = a 1 x 1 + ... + a k - 1 x k - 1 then a 1 x 1 + ... + a k - 1 x k - 1 - x k = 0, and { x 1 ,...,x n } is linearly dependent. b) Suppose Char R = 2. Then ( x 1 + x 2 ) + ( x 2 + x 3 ) + ( x 3 + x 1 ) = 2( x 1 + x 2 + x 3 ) = 0 and { x 1 + x 2 ,x 2 + x 3 ,x 3 + x 1 } is linearly dependent = . Suppose Char R 6 = 2 and a 1 ( x 1 + x 2 )+ a 2 ( x 2 + x 3 )+ a 3 ( x 3 + x 1 ) = 0. Collecting terms, ( a 1 + a 3 ) x 1 + ( a 1 + a 2 ) x 2 + ( a 2 + a 3 ) x 3 = 0 By independence of { x 1 ,x 2 ,x 3 } , we have the system of equations a 1 + a 3 = 0 a 1 + a 2 = 0 a 2 + a 3 = 0 Subtract equations two at a time to get a 1 = a 2 = a 3 . Since R is a division ring with Char
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Homework 4 - Aaron Goldsmith Homework 3 Section 4.2 Algebra...

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