Answer
See below
Work Step by Step
Given:
$x_1(t)=\begin{bmatrix}
\sin^2 t\\
\cos ^2t\\
2\\
\end{bmatrix}$
$x_2(t)=\begin{bmatrix}
2\cos^2 t\\
2\sin^2t\\
1
\end{bmatrix}$
$x_3(t)=\begin{bmatrix}
2\\
2\\
5
\end{bmatrix}$
We can see
$2x_1(t)+x_2(t)=2\begin{bmatrix}
\sin^2 t\\
\cos ^2t\\
2\\
\end{bmatrix}+\begin{bmatrix}
2\cos^2 t\\
2\sin^2t\\
1\\
\end{bmatrix}=\begin{bmatrix}
2\\
2\\
5\\
\end{bmatrix}=x_3(t)$
Hence, $x_1(t),x_2(t)$ and $x_3(t)$ are linearly independent on $(-\infty,\infty)$
