Chapter 5.5, Problem 8E

In Exercises $8-13$, a subset $S$ of vector space $V$ is given. In each case choose one of the statements $i),ii),$ or $iii)$ that holds for $S$ and verify that this is the case.

$i)$ $S$ is a basis for $V$.

$ii)$ $S$ does not span $V$.

$iii)$ $S$ is linearly dependent.

8. $S=\left\{1+x-x^2,x+x^3,-x^2+x^3\right\};V=P_3$