Answer
See below
Work Step by Step
Given:
$x_1(t)=\begin{bmatrix}
t\\
t
\end{bmatrix}$
$x_2(t)=\begin{bmatrix}
|t|\\
t
\end{bmatrix}$
Obtain:
$W_{[x_1,x_2,x_3]}=\begin{vmatrix}
t & |t|\\
t & t
\end{vmatrix}$
If we take $W_{[x_1,x_2]}(-1)=\begin{vmatrix}
-1 & |-1| \\
-1 & -1
\end{vmatrix}=\begin{vmatrix}
-1 & 1\\
-1 & -1
\end{vmatrix}=1-(-1)=2 \ne 0$ for all $t \in (-\infty, \infty)$
Hence, $x_1(t)$ and $x_2(t)$ are linearly independent on $(-\infty,\infty)$
