My professor wrote the solution to a system as
$$X = C_1 \begin{bmatrix}1 \\2 \end{bmatrix} e^{\lambda_1t} + C_2 \begin{bmatrix}3 \\4 \end{bmatrix} e^{\lambda_2t}$$
Where the column vectors are the basis to the eigenspace of the coefficient matrix and $\lambda_1$ and $\lambda_2$ are eigenvalues.
My question is why is it important to have the eigenvectors expressed in the solution? I thought all that we cared about was the eigenfunctions because those are what form our solution space?
I would've left the answer as
$$X=\mathbf{C_1}e^{\lambda_1t}+\mathbf{C_2}e^{\lambda_2t}$$
Since isn't all that we're doing anyways is finding the constant to multiply the basis vectors by? Is my way correct as well?
btw if it wasn't clear before
$$\mathbf{C_1} = \begin{bmatrix}C_{1 1} \\C_{1 2} \end{bmatrix}, \: \mathbf{C_2} = \begin{bmatrix}C_{2 1} \\C_{2 2} \end{bmatrix}$$