Consider the vector space $P_2$ and two of its subspaces, given by $S=$span$\{x^2+1,x^2-1,2x-1\}$ and $T=$span$\{1+x,1-x,1-x^2\}.$ Prove that the space $S$ is the same as $T$ by showing that span$\{x^2+1,x^2-1,2x-1\}=$span$\{1+x,1-x,1-x^2\}.$