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Unformatted text preview: MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 27 Linear Independence (contd) Example. Let V = M 22 be the real vector space corresponding to the 2 2 matrices. Consider the following three vectors: v 1 = 2 1 0 1 , v 2 = 1 2 1 0 , and v 3 = 3 2 1 . We determine whether these vectors are linearly independent. We consider the equation a 1 v 1 + a 2 v 2 + a 3 v 3 = = 2 a 1 + a 2 a 1 + 2 a 2 3 a 3 a 2 2 a 3 a 1 + a 3 = 0 0 0 0 . This corresponds to homogeneous linear system 2 a 1 + a 2 = 0 a 1 + 2 a 2 3 a 3 = 0 a 2 2 a 3 = 0 a 1 + a 3 = 0 The augmented matrix and its reduced row echelon form are 2 1 1 2 3 0 1 2 1 0 1 = 1 0 1 0 1 2 0 0 0 0 This corresponds to the system a 1 + a 3 = 0 a 2 2 a 3 = 0 = a 1 a 2 a 3 =  r 2 r r . Since there are infinitely many solutions, there is at least one nontrivial solution. In particular, when r = 1 we have the linear combination ( 1) 2 1 0 1 + (2) 1 2 1 0 + (1) 3 2 1 = 0 0 0 0 . This shows that the vectors are not linearly independent. More Criteria For Determining Linear Independence. Let ( V, + , ) be a real vector space, and consider a set S = { v 1 , v 2 , ..., v k } of k vectors in V . We focus on the case where V = R n . We can form an n k matrix where the vectors are its columns: A = a 11 a 12 a 1 k a 21 a 22 a 2 k ....
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.
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