Answer
See answer below
Work Step by Step
We are given:
$f_1(x)=1$
$f_2(x)=x$
$f_3(x)=x^2$
and $I=(-\infty, +\infty)$
The augmented matrix of this system is:
$W[f_1,f_2,f_3](x)\begin{bmatrix}
1 & x &x^2\\
0& 1 & 2x \\
0 & 0 &2
\end{bmatrix} =2 \begin{bmatrix}
1 & x\\
0 & 1
\end{bmatrix}=2$ for all $x$ in $(-\infty, +\infty)$
Hence, the set of vectors $\{f_1, f_2, f_3 \}$ is linearly independent on $(-\infty, +\infty)$
