Unformatted text preview: + x 2 n . Use the CauchySchwarz nequlity to prove that the length has the following properties. (a) k x k ≥ 0. (b) k x k = 0 if and only if ~x = ~ 0. (c) k c~x k =  c k ~x k . (d) k ~x + ~ y k ≤ k ~x k + k ~ y k . 6. Any function l from l : R n → R such that NN l ( ~x ) ≥ 0, l ( ~x ) = 0 ND If l ( ~x ) = 0, then ~x = ~ H l ( c~x ) =  c  ~x T l ( ~x + ~ y ) ≤ l ( ~x ) + l ( ~ y ) is called a norm . Show that the following functions on R n are norms. (a) l ( ~x ) = max i =1 ,...,n  x i  (b) l ( ~x ) =  x 1  + ··· +  x n  7. A reasonable distance function d should satisfy the following properties (a) d ( ~x,~ y ) = 0 if and only if ~x = ~ y . (b) d ( ~x,~ y ) ≥ 0 for vecx,~ y ∈ R n . (c) d ( ~x,~ y ) ≤ d ( ~x,~ z ) + d ( ~ z,~ y ), for ~x,~ y,~ z ∈ R n . If l is a norm, show that d ( ~x,~ y ) = l ( ~x~ y ) deﬁnes a distance function. 1...
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 Fall '08
 STAPLES
 Linear Algebra, Algebra, Trigraph

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