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11hw3 - Homework Assignment#3 11.14 Which of the following...

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Homework Assignment #3 11.14 Which of the following are bases of R 3 ? a ) 1 1 1 , 1 2 1 ; b ) 1 1 1 , 1 2 1 , 1 0 1 ; c ) 6 3 9 , 5 2 8 , 4 1 7 ; d ) 1 1 1 , 1 2 1 , 1 0 0 ; e ) 1 1 1 , 1 2 1 , 1 0 0 , 0 1 0 . Answer: a ) Not a basis. Two vectors cannot span R 3 (Thm. 11.6). b ) Not a basis. Label the vectors v 1 , v 2 , and v 3 . The vectors are linearly dependent since 2 v 1 - v 2 - v 3 = 0 . c ) Not a basis. Form the matrix whose columns are the three vectors. Its determinant is 0. Since the matrix is not invertible, the vectors do not form a basis (Thm. 11.8). d ) It is a basis. Form the matrix whose columns are the three vectors. Its determinant is - 1. Since the matrix is invertible, the vectors form a basis (Thm. 11.8). e ) Not a basis. Four vectors in R 3 must be linearly dependent (Thm. 11.3).
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