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Unformatted text preview: CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b; Solutions to Homework Set 1 Due: January 14, 2008 1. We must show W is nonempty and closed. By Remark 2 in class, 0 U for each U U , so 0 W . Thus W 6 = . Let x,y W and a,b F . Then x,y U for all U U , so as U is a subspace, ax + by U . As this holds for all U U , ax + by W , so indeed W is closed. 2. The system F is homogeneous if b i = 0 for all i . Claim S = S ( F ) is a subspace iff F is homogeneous. First if S is a subspace then by Remark 2 in Chapter 1, 0 S . But 0 = (0 ,... , 0), so b i = j a i,j 0 = j 0 = 0 as a 0 = 0 for all a F by 1.3 and 0 + 0 = 0 by definition of 0. Thus if S is a subspace then F is homogeneous. Conversely assume F is homogeneous. Let S i be the set of solutions to the i th equa- tion. We will show each S i is a subspace of V n . Then as S = T i S i , S is a subspace by Problem 1....
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