Find the general solution of $$y'=Ay,$$ where $A$ is the constant matrix
$$A=\begin{bmatrix}5&2&-3\\4&5&-4\\6&4&-4\end{bmatrix}$$
So, after determining the eigenvalues to be $\lambda=1,2,3$, I was able to find the corresponding eigenvectors for each eigenvalue to be
$$u=\begin{bmatrix}.5\\.5\\1\end{bmatrix}$$ $$v=\begin{bmatrix}1\\0\\1\end{bmatrix}$$ $$w=\begin{bmatrix}.5\\1\\1\end{bmatrix}$$
then would the general solution be
$$y(t) = C_1e^tu + C_2e^{2t}v + C_3e^{3t}w$$
where $u,v$,and $w$ are the eigenvectors defined previously?
Since there is no initial condition given then the $3$ constants cannot be determined, correct? Thus, we leave them as they are?