section5

section5 - Exercise 4. You are marking EE221A homework when...

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EE221A Section 5 9/23/11 1 Norms Exercise 1. Prove that x R n , k x k ≤ k x k 1 n k x k Exercise 2. In R 2 , sketch the unit sphere B = n x : k x k p = 1 o for p = 1 , p = 2 , p = . What about 0 < p < 1 ? 2 Complete (Banach) Spaces Exercise 3. Let X be the space of real-valued continuous functions on [0 , 1] with norm: k f k X = ˆ 1 0 | f ( t ) | dt. a) Show that X is a normed vector space. b) Use the following sequence to argue that X is not a Banach space (complete normed vector space): f n ( t ) = 0 , 0 t 1 2 - 1 n nt - n 2 + 1 , 1 2 - 1 n t 1 2 1 , 1 2 t. c) Unlike X , C [0 , 1] , with norm k x k C = max t [0 , 1] | x ( t ) | , is a Banach space. Discuss why your argument in part (b) is no longer valid when you use the norm 3 Syntax Errors
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Unformatted text preview: Exercise 4. You are marking EE221A homework when you come across the following statements in the students answers. What is wrong here? a) dim S = n-rank ( N ( A )) b) R ( A ) = r c) { v i } l i =1 R ( A ) d) A R m n , B R n p , .... v i = Bu i , .... v i R ( A ) . e) . .. the vector space U \ N ( A ) ... f) N ( A ( u )) := { u : A ( u ) = V } g) A R m n ... the case when det A = 0 ... h) dim { v 1 ,v 2 ,...,v n } = n i) N ( AB ) = k + null ( B ) 1...
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