Answer
See below
Work Step by Step
Given:
$x_1(t)=\begin{bmatrix}
e^t\\
-e^t
\end{bmatrix}$
and $x_2(t)=\begin{bmatrix}
e^t\\
e^t
\end{bmatrix}$
Obtain:
$W_{[x_1,x_2]}=\begin{vmatrix}
e^t & e^t\\
-e^t & e^t
\end{vmatrix}=e^{2t}(-e^{2t})=2e^{2t}$
Since $2e^{2t}\ne 0$ for all $t \in (-\infty, \infty)$
Hence, $x_1(t)$ and $x_2(t)$ are linearly independent on $(-\infty,\infty)$
