Basis signals for the discrete-time Fourier Series
The complex exponentials

$$\phi_k[n]=e^{j\frac{2\pi k}{N}n},\qquad k=0,1,\ldots, N-1\tag{1}$$

form an orthogonal basis, i.e.

$$\sum_{n=0}^{N-1}\phi_k[n]\phi_l^*[n]=0,\qquad k\neq l\tag{2}$$

So clearly they are all different, and they are even orthogonal (and hence not linearly dependent).

It's straightforward to show that Eq. $(2)$ is true:

$$\sum_{n=0}^{N-1}\phi_k[n]\phi_l^*[n]=\sum_{n=0}^{N-1}e^{j\frac{2\pi (k-l)}{N}n}=
\begin{cases}
\sum_{n=0}^{N-1}e^{j\frac{2\pi (0)}{N}n}=\sum_{n=0}^{N-1}1=N,\qquad &amp; k = l \\
\\
\frac{1-e^{j2\pi (k-l)}}{1-e^{j\frac{2\pi (k-l)}{N}}}=\frac{1-1}{1-e^{j\frac{2\pi (k-l)}{N}}}=0,\qquad &amp; k\neq l \\
\end{cases}\tag{3}$$