# Week05s - Math 43 Fall 2007 B Dodson Week 5 Spanning Independence and Bases Is the vector b =[3 3 4 a linear combination of v1 =[1-1 2 v2 =[2 1 3

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Math 43, Fall 2007 B. Dodson Week 5: Spanning, Independence and Bases Is the vector ± b = [3 , 3 , 4] a linear combination of ±v 1 = [1 , - 1 , 2] ,±v 2 = [2 , 1 , 3]? Solution We have 3 3 4 = c 1 1 - 1 2 + c 2 2 1 3 = 1 2 - 1 1 2 3 c 1 c 2 , where we use matrix notation from 3.1. We can easily see the system of equations directly, without using this matrix product, but writing the vector equation as the matrix equation c = ± b gives the augmented matrix of the system directly as [ A | ± b ] .

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2 Observe that this is the linear system with augmented matrix [ ±v 1 ±v 2 | ± b ] = 1 2 | 3 - 1 1 | 3 2 3 | 4 . Row reduction gives 1 2 | 3 0 3 | 6 0 - 1 | - 2 . We see that this is a consistent system. (why?) So YES, ± b is a linear combination. We are often asked to continue by ﬁnding an explicit linear combination, solving for c 1 , c 2 .
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## This note was uploaded on 02/29/2008 for the course MATH 43 taught by Professor Dodson during the Spring '08 term at Lehigh University .

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Week05s - Math 43 Fall 2007 B Dodson Week 5 Spanning Independence and Bases Is the vector b =[3 3 4 a linear combination of v1 =[1-1 2 v2 =[2 1 3

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