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

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 02/20/2010 for the course AS a taught by Professor 11 during the Spring '10 term at École Normale Supérieure.

Ask a homework question - tutors are online