Unformatted text preview: , v k , w k +1 , w k +2 , ··· , w n } is a basis for R n . c 1 v 1 + c 2 v 2 + ··· + c k v k + c k +1 w k +1 + c k +2 w k +2 + ··· + c n w n = 0 ⇒ c 1 v 1 + c 2 v 2 + ··· + c k v k =c k +1 w k +1c k +2 w k +2 ··· c n w n and span ( W ) ⊥ span ( V ) ⇒ span ( W ) ∩ span ( V ) = 0 ⇒ c 1 v 1 + c 2 v 2 + ··· + c k v k =c k +1 w k +1c k +2 w k +2 ··· c n w n = 0 ⇒ c i = 0 for 1 ≤ i ≤ n ⇒ V ∪ W is a linearly independent set. dim ( span ( W ∩ V )) = nk + k = n (= dimR n ) and span ( W ∪ V ) ⊂ R n ⇒ R n = span ( W ∪ V ) Nov. 13, 2007 Typeset by L A T E X...
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 Spring '08
 anony
 Linear Algebra, linearly independent set, 1 wk, 2 wk, rn rn

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