Let n ∈ N denote a (fixed) dimension. Suppose that $A$ is an $n\times n$ matrix with eigenvalues $λ_1,\dots, λ_n$ and corresponding eigenvectors $v_1,\dots, v_n$. Additionally, suppose that $v_1,\dots, v_n$ are distinct (linearly independent). Where $α_1,\dots, α_n$ are arbitrary constants, show that: $$y(t)=\sum_{i=1}^n \alpha_i v_i e^{{\lambda_i}t}$$ is a solution to the ODE $$y'=Ay$$
I feel like this is not a very difficult question and thus this is what I have done so far:
$$\frac{dy}{dt}=Ay$$ $$\int\frac{1}{Ay}dy=\int{dt}$$ $$\ln y=At$$ $$y=\exp(At)$$
and thus A represents the summation part of the given solution.
Am I going in the right direction??