Answer
See answer below
Work Step by Step
Assume that:
$p_1(x)=1+x$
$p_2(x)=-x+x^2$
$p_3(x)=1+2x^2$
Then we have:
$c_1p_1(x)+c_2p_2(x)+c_3p_3(x)=0$
$c_1(1+x)+c_2(-x+x^2)+c_3(1+2x^2)=0$
We obtain:
$\begin{bmatrix}
1 & 0 & 1\\
1 & -1 & 0 \\
0 & 1 & 2
\end{bmatrix} \approx\begin{bmatrix}
1 & 0 & 1\\
0 & 1 & 1 \\
0 & 1 & 2
\end{bmatrix} \approx \begin{bmatrix}
1 & 0 & 1\\
0 & 1 & 1 \\
0 & 0 & 1
\end{bmatrix}$
$\rightarrow C_1+C_3 = 0$
$C_2+C_3=0$
$C_3=0$
$\rightarrow C_1=C_2=C_3=0$
The set of vectors $\{p_1,p_2,p_3\}$ is a basis for $P_2$
