Let
$x^{(1)}(t) = \begin{bmatrix} e^{-t} \\ -4e^{-t} \end{bmatrix}$, $x^{(2)}(t) = \begin{bmatrix} 0 \\ -3e^{-t} \end{bmatrix}$, $x^{(3)}(t) = \begin{bmatrix} -3e^{-t} \\ 3e^{-t} \end{bmatrix}$
Are the vectors $x^{(1)}(t)$, $x^{(2)}(t)$ and $x^{(3)}(t)$ linearly independent?
choose
If the vectors are independent, enter zero in every answer blank since those are only the values that make the equation below true. If they are dependent, find numbers, not
all zero, that make the equation below true. You should be able to explain and justify your answer.
$\begin{bmatrix} 0 \\ 0 \end{bmatrix} = \boxed{} \begin{bmatrix} e^{-t} \\ -4e^{-t} \end{bmatrix} + \boxed{} \begin{bmatrix} 0 \\ -3e^{-t} \end{bmatrix} + \boxed{} \begin{bmatrix} -3e^{-t} \\ 3e^{-t} \end{bmatrix}$
