Answer
See below
Work Step by Step
Given:
$x_1(t)=\begin{bmatrix}
t+1\\
t-1\\
2t
\end{bmatrix}$
$x_2(t)=\begin{bmatrix}
e^t\\
e^{2t}\\
e^{3t}
\end{bmatrix}$
$x_3(t)=\begin{bmatrix}
1\\
\sin t\\
\cos t
\end{bmatrix}$
Obtain:
$W_{[x_1,x_2,x_3]}=\begin{vmatrix}
t+1 & e^t & 1\\
t-1 & e^{2t} \sin t\\
2t & e^{3t} & \cos t
\end{vmatrix}=t^3-t^2$
If we take $W_{[x_1,x_2]}(0)=\begin{vmatrix}
1 & 1 & 1\\
-1 & 1 & 0\\
0 & 1 & 1
\end{vmatrix}=\begin{vmatrix}
1 & 0\\
1 & 1
\end{vmatrix}-\begin{vmatrix}
-1 & 0\\
0 & 1
\end{vmatrix}+\begin{vmatrix}
-1 & 1 \\
0 & 1
\end{vmatrix}=1+1-1=1 \ne 0$ for all $t \in (-\infty, \infty)$
Hence, $x_1(t), x_2(t)$ and $x_3(t)$ are linearly independent on $(-\infty,\infty)$
