Unfortunately, you need to be careful, as this is false as stated, unless you allow the chart to be only Lipschitz. Consider, for example, my answer to this question, where a smooth example in dimension $2$ with $a_1=1$ and $a_2=2$ is given that cannot be linearized by a smooth (or even $C^2$) change of coordinates.
For a proof of the linearization result, you can look in any good book on dynamical systems, or consult the original paper of Hartman (Proc. Amer. Math. Soc. 11 (1960), 610-620).
