Are the real parts of the vectors constituting the discrete Fourier transform matrix linearly independent? Let W denote the n- dimensional symmetric discrete Fourier transform matrix and $W_{i}$ denote its column vectors. Then, is the set { Re($W_{i}$) | i= 1... n } linearly independent? Or similarly, find det( Re ( W ) ).

The short answer is: No.
For example, consider $n=3$. You have $\omega=e^{-2\pi i/ 3} = -\frac{1}{2} - i \frac{\sqrt{3} }{2}$.
Then; the corresponging DTF matrix, $W$, becomes 
 $$ W=  
 \begin{pmatrix}
 1 & 1 & 1 \\
 1 & \omega &\omega^2 \\
 1 & \omega^2 & \omega^4\\
 \end{pmatrix} 
=  
\begin{pmatrix}
 1 & 1 & 1 \\
 1 & e^{-2\pi i/ 3} &e^{-4\pi i/ 3} \\
 1 & e^{-4\pi i/ 3} & e^{-8\pi i/ 3}\\
 \end{pmatrix}
=
\begin{pmatrix}
 1 & 1 & 1 \\
 1 & -\frac{1}{2} - i \frac{\sqrt{3}}{2} &-\frac{1}{2} + i \frac{\sqrt{3}}{2} \\
 1 & -\frac{1}{2} + i \frac{\sqrt{3}}{2} & -\frac{1}{2} - i \frac{\sqrt{3}}{2}\\
 \end{pmatrix}.
$$
Giving: $$\Re(W)= 
\begin{pmatrix}
 1 & 1 & 1 \\
 1 & -1/2 &-1/2 \\
 1 & -1/2 & -1/2\\
 \end{pmatrix}. $$
And the set $ \{ \, \Re(W_i) \mid i= 1,2,3 \, \}
 =
\Bigg\{
\begin{pmatrix}
 1 \\
 1 \\
 1 \\
 \end{pmatrix}, 
\begin{pmatrix}
 1 \\
 -1/2 \\
 -1/2 \\
 \end{pmatrix},
\begin{pmatrix}
 1 \\
 -1/2 \\
 -1/2 \\
 \end{pmatrix}
\Bigg\}
$
is not linearly independent since the last two vectors are equal.
Moreover; we can prove that for each $n\geq3$, the column vectors of the real part of the corresponding DFT matrix $W$ are in fact linearly dependent! A simple proof is as follows:
$$ \Re(W)_2 =\Re(W_2) = \Re \Big(\begin{pmatrix}
 1 \\
 \omega \\
 \omega^2 \\
 \vdots\\
\omega^{n-1}
 \end{pmatrix}\Big) = \Re \Big(\begin{pmatrix}
 1 \\
 \omega^{-1} \\
 \omega^{-2} \\
 \vdots\\
\omega^{-(n-1)}
 \end{pmatrix}\Big) = \Re \Big(\begin{pmatrix}
 1 \\
 \omega^{-1} \times \omega^n\\
 \omega^{-2} \times \omega^{2n} \\
 \vdots\\
\omega^{-(n-1)} \times \omega^{(n-1)n}
 \end{pmatrix} \Big)  = \Re \Big(\begin{pmatrix}
 1 \\
 \omega^{n-1} \\
 \omega^{2(n-1)} \\
 \vdots\\
\omega^{(n-1)(n-1)}
 \end{pmatrix} \Big) =\Re (W_n) =\Re (W)_n
$$ 
since $\omega^{nk}=1$ for any integer $k$. 
Thus, the second and n-th columns of $\Re(W)$ are equal if $n\geq3$; which, in the end shows that the column vectors are not linearly independent.
