section2 - 9/2/11 EE221A Section 2 1 Fields 1. Show that...

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EE221A Section 2 9/2/11 1 Fields 1. Show that the set { 0 , 1 } , with multiplication defined as binary AND and addition defined as binary XOR, is a field. × (AND) 0 1 0 0 0 1 0 1 + (XOR) 0 1 0 0 1 1 1 0 2. Show that α F , α · 0 = 0 · α = 0 . 2 Vector Spaces 1. Does C form a vector space over R ? Does R form a vector space over C ? 2. Show that a set W V is a subspace of V iff α,β F , x,y W , αx + βy W . 3. Show that the additive identity θ V of a vector space V is unique. 4. Consider V = ( C ([0 , 1] , R ) , R ) , the vector space of continuous, real-valued func- tions on the interval [0 , 1] , over the reals. Prove that D := ± f ( · ) V : ˆ 1 0 f ( τ ) = 0 ² is a subspace of V . 1
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3 Linear independence 2 3 Linear independence 1. Show that the set of polynomials { 2 x - 1 ,x 2 + 2 x - 2 , 3 } is linearly independent in R [ x ] , the set of polynomials with coefficients in R . 2. Show that { e t ,e 2 t } is linearly independent in V = ( C ([0 , 1] , R ) , R ) . 4 Basis and Coordinate Representation [last 3 pages of Lecture Notes 2] Basis: B = { b 1 ,b 2 ,...,b n } is a basis of V if (i) B spans V ; (ii) B is linearly independent
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section2 - 9/2/11 EE221A Section 2 1 Fields 1. Show that...

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