Question: Do the polynomials , , , span ?
My attempt: Let be an arbitrary vector in , then:
Combining the terms on the left side and equating corresponding coefficients gives the linear system:
Which becomes the augmented matrix:
Row reducing it, I come to the matrix:
Since there are a row of zeros, then must be equal to zero for the system to be consistent, and hence, the polynomials do not span . However, my book concluded that the reason that the polynomials do not span is because it is an inconsistent system, that no value for a, b, c, and d would yield 0 for the 4th row? How did they come to this conclusion? Thanks in advance.