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# problem1 - U 1 W = U 2 W then U 1 = U 2 Problem 5 Subspaces...

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EE221A Linear System Theory Problem Set 1 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2010 Issued 9/2; Due 9/10 Problem 1: Functions. Consider f : R 3 R 3 , defined as f ( x ) = Ax,A = 1 0 0 0 1 0 0 0 0 ,x R 3 Is f a function? Is it injective? Is it surjective? Justify your answers. Problem 2: Fields. (a) Use the axioms of the field to show that, in any field, the additive identity and the multiplicative identity are unique. (b) Is GL n , the set of all n × n nonsingular matrices, a field? Justify your answer. Problem 3: Vector Spaces. Show that the set of all polynomials in s of degree k or less with real coefficients is a vector space over the field R . Find a basis. What is the dimension of the vector space? Problem 4: Subspaces. Suppose U 1 ,U 2 ,...,U m are subspaces of a vector space V . The sum of U 1 ,U 2 ,...,U m , denoted U 1 + U 2 + ... + U m , is defined to be the set of all possible sums of elements of U 1 ,U 2 ,...,U m : U 1 + U 2 + ... + U m = { u 1 + u 2 + ... + u m : u 1 U 1 ,...,u m U m } (a) Is U 1 + U 2 + ... + U m a subspace of V ? (b) Prove or give a counterexample: if U 1 ,U 2 ,W are subspaces of V such that
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Unformatted text preview: U 1 + W = U 2 + W , then U 1 = U 2 . Problem 5: Subspaces. Consider the space F oF all Functions f : R + → R , which have a Laplace transForm ˆ f ( s ) = i ∞ f ( t ) e-st dt defned For all Re ( s ) > 0. ²or some fxed s in the right halF plane, is { f | ˆ f ( s ) = 0 } a subspace oF F ? Problem 6: Linear Independence. Which oF the Following sets are linearly independent in R 3 ? 1 2 , 2 1 , 5 1 , 4 5 1 , 1 2-1 , 2 1 3 Problem 7: Bases. Let U be the subspace oF R 5 defned by U = { [ x 1 ,x 2 ,...,x 5 ] T ∈ R 5 : x 1 = 3 x 2 and x 3 = 7 x 4 } ²ind a basis For U . Problem 8: Bases. Prove that iF { v 1 ,v 2 ,...v n } is linearly independent in V , then so is the set { v 1-v 2 ,v 2-v 3 ,...,v n-1-v n ,v n } . 1...
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