rec4 - EE 221 Linear Systems Recitation#4 — Sam Burden...

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Unformatted text preview: EE 221 : Linear Systems Recitation #4 — Sam Burden — Sep 17, 2010 1 Continuity & Linearity Revisited Fact L p ( I ) := f : I → R | R I | f | p < ∞ is a (pseudo-)metric space with d ( f,g ) := (R I | f- g | p ) 1 /p . Exercise Consider A : ( L p ([0 , 1]) ,d ) → ( R , |·| ) defined by A ( f ) := f (1). (a) Is A linear? Yes: A ( a ( f + g )) = a A ( f ) + a A ( g ). (b) Is A continuous? No: x n → 0, but (1) n = 1 for all n . Eigenstructure Definition λ ∈ C is an eigenvalue of A ∈ C n × n if there is an eigenvector v 6 = 0 : Av = λv . Definition The spectrum of A is the set of eigenvalues, denoted spec A . Exercise Suppose for A : ( F n ,F ) → ( F n ,F ) there is λ ∈ F and a basis { b j } n j =1 for which A b 1 = λb 1 and A b k = λb k + b k- 1 for k = 2 ,...,n . Find the matrix representation of A . Induced Norm Let A : ( U,F, |·| U ) → ( V,F, |·| V ) be linear....
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• Fall '10
• ClaireTomlin
• Eigenvalue, eigenvector and eigenspace, LP, Compact space, Sam Burden, Linear Systems Recitation, A. Exercise Suppose

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rec4 - EE 221 Linear Systems Recitation#4 — Sam Burden...

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