Chapter 5.4, Problem 22E

22. In $P_2$, let $S=\left\{p_1\left(x\right),p_2\left(x\right),p_3\left(x\right),p_4\left(x\right)\right\}$, where $p_1\left(x\right)=1+2x+x^2$ $p_2\left(x\right)=2+5x$ $p_3\left(x\right)=3+7x+x^2$ and $p_4\left(x\right)=1+x+3x^2$.Use the method illustrated in Example $8$ to obtain a basis for $Sp\left(S\right)$. [ Hint: Use the basis $B=\left\{1,x,x^2\right\}$ to obtain coordinate vectors for $p_1\left(x\right),p_2\left(x\right),p_3\left(x\right)$ and $p_4\left(x\right)$. Now use the method illustrated in Example $7$ of section $3.4$.]