08aAss3 - AS3/MATH1111/YKL/08-09 THE UNIVERSITY OF HONG...

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AS3/MATH1111/YKL/08-09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 3 Due date : Nov 3, 2008 before 6:30 p.m. Where to hand-in : Assignment Box outside the lifts on the 4th floor of Run Run Shaw Remember to write down your Name , Uni. no. and Tutorial Group number . 1. Let V be a vector space. (a) Let W be a nonempty subset of V . The span of W , denoted by Span( W ), is defined as the set of all possible linear combinations of elements in W . i.e. Span( W ) = { a 1 w 1 + a 2 w 2 + ··· + a r w r : a 1 , ··· ,a r R ,w 1 , ··· w r W, r N } . (Here N = { 1 , 2 , ···} is the set of natural numbers.) Show that W = Span( W ) if and only if W is a subspace. (b) Suppose S V (i.e. S is a subset of V ). A subspace U of V is said to be the smallest subspace containing S if U satisfies the following property: U W whenever W is a subspace of V and S W . Suppose
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08aAss3 - AS3/MATH1111/YKL/08-09 THE UNIVERSITY OF HONG...

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