Answer
See answer below
Work Step by Step
We are given:
$f_1(x)=e^{2x}$
$f_2(x)=e^{3x}$
$f_3(x)=e^{-x}$
and $I=(-\infty, +\infty)$
The augmented matrix of this system is:
$W[f_1,f_2,f_3](x)=\begin{bmatrix}
e^{2x} & e^{3x} & e^{-x}\\
2e^{2x} & 3e^{3x} & -e^{-x} \\
4e^{2x} & 9e^{3x} & e^{-x}
\end{bmatrix}=e^{2x}\begin{bmatrix}
3e^{3x}& -e^{-x} \\
9e^{3x} & e^{-x}
\end{bmatrix}-e^{3x}\begin{bmatrix}
2e^{2x}& -e^{-x} \\
4e^{2x} & e^{-x}
\end{bmatrix}+e^{-x}\begin{bmatrix}
2e^{2x}& 3e^{3x} \\
4e^{2x} & 9e^{3x}
\end{bmatrix}=e^{2x}[3e^{2x}-(-9e^{2x})]-e^{3x}[2e^{x}-(-4e^{x})]+e^{-x}(18e^{5x}-12e^{5x})=12e^{4x}$ for all $x$ in $(-\infty, \infty)$
Hence, the set of vectors $\{f_1, f_2, f_3 \}$ is linearly independent on $(-\infty, +\infty)$
