The unique solution of the linear ordinary differential equation system $$\frac{d}{dt}\mathbf{y}(t) = \mathbf{A} \mathbf {y}(t)$$ is supposedly given by $$\mathbf{y}(t) = (\sum_{k=0}^{\infty} \frac{1}{k!}(\mathbf{A}t)^k)\mathbf{y}(0)$$ where $\mathbf{A}$ is a real or complex matrix (cf. linear differential equations system).
How to derive this rigorously (and not just prove it by plugging it in the original differential equation)?
From the given equation,
$$\frac{d^2}{dt^2}\mathbf y(t)=\frac{d}{dt}\mathbf A\mathbf y(t)=\mathbf A\frac{d}{dt}\mathbf y(t)=\mathbf A^2\mathbf y(t)$$
and by induction $$\frac{d^n}{dt^n}\mathbf y(t)=\mathbf A^n\mathbf y(t).$$
Then by a simple application of Taylor around $t=0$,
$$\mathbf y(t)=\sum_{n=0}^\infty\left.\frac{d^n}{dt^n}\mathbf y(t)\right|_{t=0}\frac{t^n}{n!}=\sum_{n=0}^\infty\mathbf A^n\mathbf y(0)\frac{t^n}{n!}=\sum_{n=0}^\infty\frac{(\mathbf At)^n}{n!}\mathbf y(0).$$
Uniqueness can probably be established using the Picard-Lindelöf theorem.