1. (a) Show that B = [1 - 3x + 7x 2 ; 4 - x 2 ; 2 + x - x 2 ] is a basis for P2.

Question

1. (a) Show that B = [1 - 3x + 7x2; 4 - x2; 2 + x - x2] is a basis for P2.

Explanation / Answer

If you let the vector be in the form for ax^2+bx+c ---> (a,b,c) Your three vectors are { (7,-3,1),(-1,0,4),(-1,1,2)} To show that it is a basis, it must span P2 and the vectors must be linearly independent. To show it is linearly independent, put the vectors into the columns of a matrix and reduce. You need to show that that matrix, A, for any vector x, satisfies the equation Ax=0. The matrix row reduces to the identity matrix so it does satisfy the equation (it has only the trivial soln x=(0,0,0)). Does it span P2? We have to check that any vector can be written as a sum of the vectors, so any polynomial in P2 must be within the column-space of A. The row-reduced form is the identity, so that means that the basis for the column-space has three vectors (dimension=3). Since the dimension of the column-space is the same as the dimension of the basis for P2 (recall {1,x,x^2} or {(1,0,0),(0,1,0),(0,0,1)} if you keep my notation from above), the column-space is also a basis for P2. So this set is both linearly independent and a spanning set for P2, so it DOES form a basis.

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