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# HW5B - Math 103B Homework Solutions HW5 Jacek Nowacki...

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Math 103B Homework Solutions HW5 Jacek Nowacki May 13, 2007 Chapter 19 Problem 2. Prove that a nonempty subset U of a vector space V over a field F is a subspace of V if, for every u and u in U and every a in F , u + u U and au U . Proof. Remark: Let’s begin by noting that if you are investigating a subset defined in some universal way (e.g. a kernel of a homomorphism), then it is necessary to check that the subset is nonempty. It does not make sense to define something and then see that it does not exist, i.e. get an empty set. OK, we begin the proof. It is implied in the problem that the operation in U is inherited from V . Hence, we do not need to check that the four conditions are satisfied – they are automatically true. A subspace is just a subgroup (under addition) that is also closed under scalar multiplication. Now, one of our subgroup tests tells us that all we have to do is show that for all x, y U (we can do this since U is not empty), x - y U . It should be clear that - y = ( - 1) · y , so that - y is in U

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HW5B - Math 103B Homework Solutions HW5 Jacek Nowacki...

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