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problem1 - EE221A Linear System Theory Problem Set 1...

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Unformatted text preview: EE221A Linear System Theory Problem Set 1 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2007 Issued 9/4; Due 9/13 Problem 1: Fields. (a) Define addition and multiplication on { , 1 } to form a field. Show that your result is a field. (b) Use the axioms of the field to show that, in any field, the additive identity and the multiplicative identity are unique. Problem 2: Vector Spaces. Let R 2 × 2 be the set of all 2 × 2 real matrices. (a) Briefly verify that R 2 × 2 is a vector space under usual matrix addition and scalar multiplication. Don’t turn this in. (b) What is the dimension of R 2 × 2 ? (c) Find a basis for R 2 × 2 . (d) Let A = bracketleftbigg 1 1 2 bracketrightbigg Is the set { I,A,A 2 } linearly dependent or independent in R 2 × 2 ? Problem 3: Subspaces. Consider the space F of all functions f : R + → R , which have a Laplace transform ˆ f ( s ) = integraltext ∞ f ( t ) e- st dt defined for all Re ( s ) > 0. For some fixed0....
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