problem43

# problem43 - February 28, 2008 If u1 , u2 and u3 are in the...

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February 28, 2008 If u 1 , u 2 and u 3 are in the span of v 1 and v 1 then u 1 , u 2 and u 3 are linearly dependent. There are two cases to consider but in both cases we simply use the deﬁn- tions. Case 1) v 1 and v 2 are linearly dependent. In this case everything can be written in terms of v 1 so everything is automatically linearly dependent. Case 2) v 1 and v 2 are linearly independent. We want to show that there exist constants k 1 , k 2 and k 3 which are not all 0 and k 1 u 1 + k 2 u 2 + k 3 u 3 = 0. Since the u i are in the span of v 1 and v 2 we can express them as linear combinations of v 1 and v 2 like so u 1 = a 1 v 1 + a 2 v 2 u 2 = b 1 v 1 + b 2 v 2 u 3 = c 1 v 1 + c 2 v 2 Now multiply each u i with k i add them together and equate the sum to 0 to get ( k 1 a 1 + k 2 b 1 + k 3 c 1 ) v 1 + ( k 1 a 2 + k 2 b 2 + k 3 c 1 ) v 2 = 0 Since we assumed that v 1 and v 2 were linearly independent we can conclude

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## This note was uploaded on 03/31/2008 for the course MATH 225 taught by Professor Guralnick during the Spring '07 term at USC.

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problem43 - February 28, 2008 If u1 , u2 and u3 are in the...

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