Unfortunately, you need to be careful, as this is false as stated, unless you allow the chart to be only $C^1$.  Consider, for example, my answer to [this question][1], 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 $C^1$ result, you can look in any good book on dynamical systems.


  [1]: http://mathoverflow.net/questions/76971/nice-metrics-for-a-morse-gradient-field-counterexample-request