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Quiz7Solutions

# Quiz7Solutions - Newberger Math 247 Spring 03 Quiz 4.1-4.2...

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Unformatted text preview: Newberger Math 247 Spring 03 Quiz 4.1-4.2 name: 1. State the deﬁnition of a subspace of Rn . A subspace of Rn is any set H in Rn that has three properties: (a) The zero vector is in H . (b) For each u and v in H , the sum u + v is in H . (c) For each u in H and each scalar c, the vector cu is in H . 2. Show that the following set is a subspace of R3 . a + 2b + c − 3d a, b, c, d ∈ R 2a + b − c H= b + c − 2d H is the set of all vectors that can be written in the form: a + 2b + c − 3d 1 2 1 −3 2a + b − c = 2 a + 1 b + −1 c + 0 d. b + c − 2d 0 1 1 −2 Thus H is the set of all linear combinations of the vectors 1 2 1 −3 2 , 1 , −1 , and 0 . 0 1 1 −2 So 2 1 −3 1 H = Span 2 , 1 , −1 , 0 , 0 1 1 −2 which means H is a subspace, since all spans are subspaces. 3. Show that the following set is a subspace of R3 . a a + 3b − c = 0 W= b b+c+a=0 c a Elements in W are vectors b that satisfy c a + 3b − c = 0 b + c + a = 0, which can be rewritten as the following matrix equation: a 1 3 −1 0 b= . 11 1 0 c Thus W is the set of all solutions to the homogeneous equation associated 1 3 −1 to the matrix A = , i.e. W = Nul A. So W is a subspace 11 1 since null spaces are subspaces. ...
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Quiz7Solutions - Newberger Math 247 Spring 03 Quiz 4.1-4.2...

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