Answer
See below
Work Step by Step
Given:
$x_1(t)=\begin{bmatrix}
t\\
t^2\\
-t^3\\
\end{bmatrix}$
$x_2(t)=\begin{bmatrix}
2t\\
3t^2\\
0
\end{bmatrix}$
$x_3(t)=\begin{bmatrix}
-t\\
0\\
3t^3
\end{bmatrix}$
We can see
$-3x_1(t)+x_2(t)=-3\begin{bmatrix}
t\\
t^2\\
-t^3\\
\end{bmatrix}+\begin{bmatrix}
2t\\
3t^2\\
0\\
\end{bmatrix}=\begin{bmatrix}
-t\\
0\\
3t^3\\
\end{bmatrix}=x_3(t)$
Hence, $x_1(t),x_2(t)$ and $x_3(t)$ are linearly independent on $(-\infty,\infty)$
