problem3

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 9/20; Due 9/28 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: Norms. Show that for x R n , || x || ≤ || x || 2 n || x || . Problem 3: Continuity and Linearity. Show that any linear map between Fnite dimensional vector spaces is continuous. Problem 4. Prove that the induced matrix norm
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