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Math 54 - Quiz 03 Ans, Spring 2006 (2)

# Math 54 - Quiz 03 Ans, Spring 2006 (2) - ± a ab ab b ² is...

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Math 54, Quiz 3 Solutions Sections 202/203 GSI Carter 1. For each of the following statements, write the word “true” or “false.” (a) The set of matrices of the form ± a a - b a + b b ² is a linear vector space. TRUE. The operations of addition and scalar multiplication have not been modiﬁed, so we only need to check that the given set is closed under those operations. We have that λ ± a a - b a + b b ² = ± λa λa - λb λa + λb λb ² and ± a a - b a + b b ² + ± c c - d c + d d ² = ± a + c ( a + c ) - ( b + d ) ( a + c ) + ( b + d ) b + d ² , so the given set of matrices is a linear vector space. (b) The set of matrices of the form
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Unformatted text preview: ± a ab ab b ² is a linear vector space. FALSE. The set is not closed under scalar multiplication, as demonstrated by 2 ± 1 1 1 1 ² = ± 2 2 2 2 ² . 2. Let v 1 = (1 , 2 , 0) and v 2 = (-1 , 3 , 1). Find perpendicular vectors p 1 and p 2 such that Span { p 1 , p 2 } = Span { v 1 , v 2 } . We use the Gram-Schmidt process. Let p 1 = v 1 and p 2 = v 2-v 2 · p 1 p 1 · p 1 p 1 = (-1 , 3 , 1)-5 5 (1 , 2 , 0) = (-2 , 1 , 1) . 1...
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