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