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Unformatted text preview: ,...,Y s are linearly dependent. So we consider the equation x 1 Y 1 + + x s Y s = 0 . Now Y i = r j =1 a ij X j , for 1 i s . Hence x 1 Y 1 + + x s Y s = x 1 ( a 11 X 1 + + a 1 r X r ) + + x r ( a s 1 X 1 + + a sr X r ) . = y 1 X 1 + + y r X r , (1) where y j = a 1 j x 1 + + a sj x s . However the homogeneous system y 1 = 0 , , y r = 0 has a nontrivial solution x 1 ,...,x s , as s > r and from (1), this results in a nontrivial solution of the equation x 1 Y 1 + + x s Y s = 0 . 39...
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 Fall '10
 Dreibelbis

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