Linear systems of ODEs

Are $\left[ \begin{smallmatrix} e^{2t} \\ e^t \end{smallmatrix}\right]$ and $\left[ \begin{smallmatrix} e^{t} \\ e^{2t} \end{smallmatrix}\right]$ linearly independent? Justify.

Are $\left[ \begin{smallmatrix} \cosh(t) \\ 1 \end{smallmatrix}\right]$, $\left[ \begin{smallmatrix} e^{t} \\ 1 \end{smallmatrix}\right]$, and $\left[ \begin{smallmatrix} e^{-t} \\ 1 \end{smallmatrix}\right]$ linearly independent? Justify.

Write $x'=3x-y+e^t$, $y'=tx$ in matrix notation.

1. Write $x_1'=2tx_2$, $x_2'=2tx_2$ in matrix notation.
2. Solve and write the solution in matrix notation.