Answer
See below
Work Step by Step
Given:
$x_1(t)=\begin{bmatrix}
\sin t\\
\cos t\\
1
\end{bmatrix}$
$x_2(t)=\begin{bmatrix}
t\\
1-t\\
1
\end{bmatrix}$
$x_3(t)=\begin{bmatrix}
\sinh t\\
\cosh t\\
1
\end{bmatrix}$
Obtain:
$W_{[x_1,x_2,x_3]}=\begin{vmatrix}
\sin t & t & \sinh t\\
\cos t & 1-t & \cosh t\\
1 & 1 & 1
\end{vmatrix}\\
=\sin t-t\sin t-t\cos t+t\sinh t+\cos t -\sinh t+t\cosh t-\sin t\cosh t$
Take $t=1 \rightarrow W_{[x_1,x_2,x_3]}(2)=6.14$
Since $6.14 \ne 0$
Hence, $x_1(t),x_2(t)$ and $x_3(t)$ are linearly independent on $(-\infty,\infty)$
