Let $T: \mathbb{R}^k / \mathbb{Z}^k \to \mathbb{R}^k / \mathbb{Z}^k$ be a linear toral endomorphism, ie.e $$T(x_1, \dots, x_k)=A(x_1, \dots ,x_k)\; \text{mod 1}$$ where $A=(a_{ij})$ is a $k\times k$ matrix with entries in $\mathbb{Z}$ with $\det A\neq 0$. Let $f \in L^1(\mathbb{R}^k / \mathbb{Z}^k, \mathscr{B}, \mu)$ be an integrable function defined on the torus, with $\mu$ the Lebesgue measure. For each $n=(n_1,\dots, n_k)\in \mathbb{Z}^k$ define $$c_n = \int f(x_1, \dots, x_k)e^{-2\pi i\langle n, x \rangle} d\mu.$$ We associate to such an $f$ its Fourier series: $$\sum_{n\in \mathbb{Z}^k} c_n e^{2\pi i\langle n,x \rangle}.$$
Then why is the Fourier series of $f\circ T$ $$\sum_{n\in \mathbb{Z}^k} c_n e^{2\pi i\langle n,Ax \rangle}.$$
I can't figure out how to obtain this Fourier series using the definition of $c_n$ and not simply by plugging in $Ax$ in place of $x$.