Answer
See answer below
Work Step by Step
We are given:
$f_1(x)=1$
$f_2(x)=3x$
$f_3(x)=x^2-1$
and $I=(-\infty, +\infty)$
The augmented matrix of this system is:
$W[f_1,f_2,f_3](x)=\begin{bmatrix}
1 & 3x & x^2-1\\
0 & 3 & 2x \\
0 & 0 & 2
\end{bmatrix}=0-0+2\begin{bmatrix}
1& 3x \\
0 & 3
\end{bmatrix}=2\times3=6$
Hence, the set of vectors $\{f_1, f_2, f_3 \}$ is linearly independent on $(-\infty, +\infty)$
