Question

I need help with both

4. Does the set S = {1 + 2x – 3x2, 1 + 5x + x?, - 2 - x + 10x?} span P2? If not, describe the set of all b that is in the span of S. All your calculations must be in matrix form, no fractions for addition or subtractions in row reductions, make sure to describe how you set up the matrix, and justify your answer.

Answer 1

3. (a). If x +3y-2z = 0, then x = -3y+2z so that (x,y,z)^T = (-3y+2z,y,z)^T = y(-3,1,0)^T +z (2,0,1)^T.

This implies that every vector in H is a linear combinationof 2 linearly independent vectors (-3,1,0)^T ,(2,0,1)^T. Hence H is a vector space and therefore a subspace of R³.

(b). H = Nul A, where A =

-3	1	0
2	0	1

so H is a subspace of R³.

4. Let A =

1	1	-2
2	5	-1
-3	1	10

It may be observed that the entries in the columns of A are the   scalar multiples of 1 and the coefficients of x,x² in the polynomials in the set S.

The RREF of A is

1	0	-3
0	1	1
0	0	0

This implies that -2+x+10x² = -3(1+2x-3x²)+( 1+5x+x²) . Thus, the polynomials in the set S are not linearly independent   . Hence dim(S) = 2. Since dim(P₂) = 3, hence the polynomials in the set S do not span P₂.

Further span (S) contains all the linear combinations of the polynomials 1+2x-3x², 1+5x+x².