problem3 - EE221A Linear System Theory Problem Set 3...

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EE221A Linear System Theory Problem Set 3 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2010 Issued 9/22; Due 9/30 Problem 1. Let A : R 3 R 3 be a linear map. Consider two bases for R 3 : E = { e 1 ,e 2 ,e 3 } of standard basis elements for R 3 , and B = 1 0 2 , 2 0 1 , 0 5 1 Now suppose that: A ( e 1 ) = 2 - 1 0 , A ( e 2 ) = 0 0 0 , A ( e 3 ) = 0 4 2 Write down the matrix representation of A with respect to (a) E and (b) B . Problem 2: Representation of a Linear Map. Let A be a linear map of the n-dimensional linear space ( V,F ) onto itself. Assume that for some λ F and basis ( v i ) n i =1 we have A v k = λv k + v k +1 k = 1 ,... ,n - 1 and A v n = λv n Obtain a representation of A with respect to this basis. Problem 3: Norms. Show that for x R n , 1 n || x || 1 ≤ || x || 2 ≤ || x || 1 . Problem 4. Prove that the induced matrix norm:
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

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