Does this set of polynomials span the vector space?

Question

Let $S_1$ = {${1 + 4x + 5x^2, 2 + 2x + 3x^2, 5 + 8x + 10x^2, 8 + 14x + 19x^2}$}

Does $S_1$ span $P_2?$

If not, find a vector in $P_2$ not in $Span(S_1)$

So far, I tried to reduce it to RREF and then saw that its rank was 3, making me think it does not span $P_2$, but if that is the case, how do I find a vector that does span $P_2$?

Answer 1

The rank is 3, that means your polynomials span a three dimensional subspace of $P_2.$

Note that$ P_2$ is three dimensional and the only three dimensional subspace of $P_2$ is $P_2$ itself.

Thus your polynomials span $P_2.$