Determine whether the given vector functions are linearly dependent or linearly independent on the interval $(-\infty, \infty)$.
$\begin{bmatrix} 2te^{-4t} \\ e^{-5t} \end{bmatrix}$, $\begin{bmatrix} e^{-t} \\ e^{-2t} \end{bmatrix}$
Let $x_1 = \begin{bmatrix} 2te^{-4t} \\ e^{-5t} \end{bmatrix}$ and $x_2 = \begin{bmatrix} e^{-t} \\ e^{-2t} \end{bmatrix}$. Select the correct choice below, and fill in the answer box to complete your choice.
A. The vector functions are linearly dependent since there exists at least one point $t$ in $I$ where $det[x_1(t) \ x_2(t)]$ is not 0. In fact, $det[x_1(t) \ x_2(t)] = \Box$.
B. The vector functions are linearly independent since there exists at least one point $t$ in $I$ where $det[x_1(t) \ x_2(t)]$ is 0. In fact, $det[x_1(t) \ x_2(t)] = \Box$.
C. The vector functions are linearly dependent since there exists at least one point $t$ in $I$ where $det[x_1(t) \ x_2(t)]$ is 0. In fact, $det[x_1(t) \ x_2(t)] = \Box$.
D. The vector functions are linearly independent since there exists at least one point $t$ in $I$ where $det[x_1(t) \ x_2(t)]$ is not 0. In fact, $det[x_1(t) \ x_2(t)] = \Box$.
