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Unformatted text preview: 1.7 Linear Independence Recall that vectory is a linear combination of vector v 1 , vector v 2 , , vector v n if vector y = c 1 vector v 1 + c 2 vector v 2 + + c n vector v n for some c 1 ,c 2 , ,c n R Q: How many ways can c 1 ,c 2 , ,c n be chosen? A: This is the soln set to a system bracketleftBig vector v 1 vector v 2 vector v n bracketrightBig c 1 c 2 . . . c n = vector y Avector c = vectory If vectory = c 1 vector v 1 + c 2 vector v 2 + + c n vector v n = d 1 vector v 1 + d 2 vector v 2 + + d n vector v n , then c 1 d 1 c 2 d 2 . . . c n d n is a soln to Avectorx = vector Definition: vector v 1 , vector v 2 , , vector v n are linearly independent if the only soln to c 1 vector v 1 + c 2 vector v 2 + + c n vector v n = vector 1 is c 1 = c 2 = = c n = 0. If there is a nontrivial soln c 1 ,c 2 , ,c n R , then we say that vector v 1 , vector v 2 , , vector v n are linearly dependent and c 1 vector v 1 + c 2 vector v 2 + + c n vector v n = vector is a linear dependency relation . EX: 1 , 1 linearly independent set of vectors EX: 1 2 1 , 1 1 , 1 4 5 is not a linearly independent set because (i) 1 4 5 = 2...
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 Spring '10
 Russel
 Linear Algebra, Algebra, Linear Independence

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