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Ma1banHw1Sol

# Ma1banHw1Sol - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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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. Let u = ( u 1 , . . . , u n ) and v = ( v 1 , . . . , v n ) be solutions to a 1 x 1 + · · · + a n x n = 0 and b, c F . Then a 1 ( bu 1 + cv 1 ) + · · · + a n ( bu n + cv n ) = b

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Ma1banHw1Sol - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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