l12 - Math 20F Linear Algebra Lecture 12:. Recap : Why do...

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Unformatted text preview: Math 20F Linear Algebra Lecture 12:. Recap : Why do we care about the null space and the column space? A x = x Nul A. A x = b is consistent b Col A. 4.5 Dimension Recall that b 1 ,..., b n form a basis for a vector space V if (i) b 1 ,..., b n are linearly independent, and (ii) b 1 ,..., b n span V . Th If { v 1 ,..., v n } is a spanning set for V then any collection of p vectors { u 1 ,..., u p } , where p > n , is linearly dependent. Pf Since v 1 ,..., v n span V we can write each u i as a linear combination: u i = a i 1 v 1 + ... + a in v n A linear combination of the u i can be written c 1 u 1 + ... + c p u p = c 1 ( a 11 v 1 + + a 1 n v n ) + + c p ( a p 1 v 1 + + a pn v n ) = ( c 1 a 11 + + c p a p 1 ) v 1 + + ( c 1 a p 1 + + c p a pn ) v n Hence c 1 u 1 + + c p u p = if c 1 a 11 + + c p a p 1 = 0 . . . c 1 a 1 n + + c p a pn = 0 Since p > n this is a homogenous system for c 1 , ,c p with more unknowns than equations so it has a nontrivial solution. Hence there are constants c 1 ,...,c p not all zero such that c 1 u 1 + + c p u p = , i.e. u 1 ,..., u...
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This note was uploaded on 04/29/2008 for the course MATH 20F taught by Professor Buss during the Spring '03 term at UCSD.

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l12 - Math 20F Linear Algebra Lecture 12:. Recap : Why do...

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