1
$\begingroup$

In one of the homeworks that I am dealing with for Linear Systems course, I have encountered with such a statement: Consider $\mathbb{C}^N$ the vector space of N dimensional complex vectors. We can define a basis $F=\{f_1,\ldots,f_N\}$ where \begin{align} f_k = \begin{bmatrix} f_{k,1} \\ f_{k,2} \\ .\\ .\\ f_{k,N} \end{bmatrix} \end{align}

and $f_{k,l} = \frac{1}{N}e^{\frac{j2\pi(k-1)(l-1)}{N}}$. It is straightforward to show that those vectors are orthogonal, but I have no idea about how to show that those vectors are linearly independent, and their span is $\mathbb{C}^N$. Could you please give me a clue?

$\endgroup$
1
$\begingroup$

Hint 1: you can rewrite products of the exponential argument:

$$ f_{k,l}=\frac{1}{N}\left(e^{\frac{j2\pi(k-1)}{N}}\right)^{l-1}$$

and recognize a Vandermonde matrix.

Hint 2: consider the dimension of the subspace spanned, and the linear independence.

$\endgroup$
  • 1
    $\begingroup$ First hint has solved all of my problems. Thank you so much:) $\endgroup$ – Canberk Oct 21 '18 at 16:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.