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).

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).

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, 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.

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).