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