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# 106 - Span of a set of vectors Properties Spancfw_u = the...

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1 Span of a set of vectors Properties: Span{ u } = the set of all multiple of u , and Span{ 0 } = { 0 }. S contains a nonzero vector. Span S has infinitely many vectors. Example: : Span S 3 = Span S 4 = R 2 nonparallel vectors Example: Span{ e 1 , e 2 } = xy -plane in R 3 Span{ e 3 } = z -axis in R 3 Example: Span S ? A = [ " ] The reduced row echelon form of [ A v ] is The reduced row echelon form of [ A w ] is v Span S w Span S

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2 Definition: If S , V R n , and Span S = V , then S is a generating set for V , or S generates V . Examples: generates R 3 ? A = [ " ] for any v in R 3 , let [ R c ] be the reduced row echelon form of [ A v ], then [ R c ] has no nonzero rows with only entries from c for any v in R 3 , v = A x for some x , thus Span S = R 3 . : rank = 3 (e) There is a pivot position in each row of A . Proof (a) (b): for any b in R m , b = A x for some x . (c)
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106 - Span of a set of vectors Properties Spancfw_u = the...

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