Unformatted text preview: is 1, x, x^2, x^3, x^4, x^5, x^6, x^7. Thus the dimension of V is 8. (5) The two columns of A are independent. The column vector b is in CS(A) if and only if the 3 by 3 augmented matrix [Ab] has dependent columns, i.e., if and only if the augmented matrix does not have 3 pivots. An echelon form for this augmented matrix has first row 2 1 b1, second row 0 5 b2+2b1, and third row 0 0 b3+(b17b2)/5. This last entry must be 0 in order to avoid having 3 pivots, i.e., b3 = (7b2  b1)/5. (6i) True: T(v2) = 1v1 + 0 v2 = v1. (6ii) True: The matrix 0 2 1 equals B^1 A B, where A is the matrix 0 1 2 and B is the matrix that takes the old basis to the new, i.e., B is 1 1 1 2 . Note that B^1 is 2 11 1 ....
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 Winter '03
 BUSS
 Linear Algebra, Algebra, Determinant

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