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19-Norms-and-Conditioning-4UP

# 19-Norms-and-Conditioning-4UP - Vector norms Norms errors...

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Vector norms Norms & errors Conditioning Norms and Conditioning Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Norms and Conditioning MATH 2070U 1 / 19 Vector norms Norms & errors Conditioning Norms and Conditioning 1 Vector norms 2 Norms, errors and linear equations 3 Conditioning of linear equations c D. Aruliah (UOIT) Norms and Conditioning MATH 2070U 2 / 19 Vector norms Norms & errors Conditioning Motivation for vector norms Real numbers are ordered : given a , b R , either a < b , a > b , or a = b Vectors in R n are not ordered,e.g., expressions like 1 - 1 1 > - 1 1 - 1 or - 1 2 - 1 < 1 - 1 3 do not make sense (at least as scalars) Norms provide a way to order vectors, measure distance c D. Aruliah (UOIT) Norms and Conditioning MATH 2070U 4 / 19 Vector norms Norms & errors Conditioning Vector norms Definition (Vector norm) Given a vector space V , a norm is a function · : V [ 0, ) satisfying three postulates: 1 v > 0 if v = 0 for every v V 2 λ v = | λ | v for every λ C , v V 3 u + v u + v for every u , v V (triangle inequality) x provides notion of length of vector x x - y provides notion of distance between vectors x , y c D. Aruliah (UOIT) Norms and Conditioning MATH 2070U 5 / 19

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Vector norms Norms & errors Conditioning The 2 -norm The 2 -norm x 2 : = n k = 1 | x k | 2 1 2 x R n Also called Euclidean norm or 2-norm Given x in workspace, use norm(x,2) in M ATLAB 3, - 4, 0, 3 2 T 2 = ( 3 ) 2 + ( - 4 ) 2 + ( 0 ) 2 + 3 2 2 = 1 2 109 [ 2, 1, -
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19-Norms-and-Conditioning-4UP - Vector norms Norms errors...

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