my first instinct is to say check your arithmetic for your row-reduction
EDIT: also, note that you are choosing a,b,c,d arbitrarily, you are not solving for them.
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.
the product on the last row in the last column should be a sum.
if your four polynomials span P3, then one linear combination (at least) must be t^3 (which is a = 1, b = c = d = 0).
this leads to 0 = -2 (from the 4th row, where your augmented column should read (d-a-2b)+(1/2)(c-2a+b)), which is inconsistent.
the zero-vector (the polynomial 0t^3 + 0t^2 + 0t + 0 in this case) is ALWAYS in the span of ANY subset.
thus setting a = b = c = d = 0 is a spectacularly bad choice for a test polynomial, because it tells us nothing (you can
always make a linear combination of 0's be 0).