hw3_solu - Solutions for Selected problems of Homework 3 A....

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Solutions for Selected problems of Homework 3 A. Agarwal 1016-331 Linear Algebra-1 Spring 2009-10 1. Q12, P.100 To show that the span is entire R 3 , we need to show that any vector b a b c B R 3 cab be expressed as a linear combination of the given vectors. In other words, we need to show that the following system is consistent for all values of a, b, and c . x 1 2 3 + y - 1 - 1 0 + z 2 1 - 1 = a b c The augmented matrix will row reduce as follows: 1 - 1 2 a 2 - 1 1 b 3 0 - 1 c some row operations later ----------------→ 1 - 1 2 a 0 1 - 3 b - 2 a 0 0 2 c + 3 a - 3 b This system has 3 variables and 3 pivots, hence it is consistent and will have a unique solution for all a, b and c . Thus every vector in R 3 is expressible as a linear combination of the given vectors. 2. Q16, P.100 Let ± b 1 b 2 b 3 ² Span of the given vectors. Then the following system has to be consistent (see theorem 2.4 ) 1 - 1 0 b 1 0 1 - 1 b 2 - 1 0 1 b 3 After performing row operations, we get 1 - 1 0 b 1 0 1 - 1 b 2 0 0 0 b 1 + b 2 + b 3 1
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For this to be consistent, we need b 1 + b 2 + b 3 = 0. Thus the span of the given vectors can be described as Span = b 1 b 2 b 3 v v v b 1 + b 2 + b 3 = 0 For a geometrical description , consider the following: b 1 b 2 b 3 = - b 2 - b 3 b 2 b 3 = b 2 - 1 1 0 + b
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This note was uploaded on 05/11/2010 for the course MTH 1016.331 taught by Professor Dr.anuragagarwal during the Spring '10 term at RIT.

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hw3_solu - Solutions for Selected problems of Homework 3 A....

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