Homework Assignment #311.14 Which of the following are bases ofR3?a)111,121;b)111,121,101;c)639,528,417;d)111,121,100;e)111,121,100,010.Answer:a) Not a basis. Two vectors cannot spanR3(Thm. 11.6).b) Not a basis. Label the vectorsv1,v2, andv3. The vectors are linearly dependent since 2v1-v2-v3=0.c) Not a basis. Form the matrix whose columns are the three vectors. Its determinant is 0. Since thematrix 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. Sincethe matrix is invertible, the vectors form a basis (Thm. 11.8).e) Not a basis. Four vectors inR3must be linearly dependent (Thm. 11.3).
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