how is DFT the change of basis operator? I understand that DFT are the coefficients when we write a vector z with respect to the Fourier basis.But the following statements are giving me a vague picture about the idea but not very clear,they are
1)the DFT is the change of basis operator that converts from euclidean basis to the  Fourier basis.
2)Fourier inversion formula is the change of basis formula for the Fourier basis. 
So how are the change of basis operator and DFT related ?
Can somebody help me with an idea.Thanks in advance.
 A: Ok, I’m probably not qualified to answer this but I will give it a shot. 
In linear algebra, you’ve probably learned how to take the dot product of two vectors. You do so by summing up the product of each respective element for both vectors. You probably also know that, given a set of linearly independent vectors, you may represent other vectors in that space as a sum of those linearly independent vectors. Doing so is known as a change of base. To find the change of base, you take the dot product of your vector you want to change with each of the linearly independent basis vectors. Each dot product will correspond to the amount of that basis vector needed for the change of base.
Now, if you look at the Fourier transform, you can see that each step actually looks like a dot product because each step is a sum of products of the input signal / vector and a complex exponential. Essentially, the complex exponentials are your basis vectors and your input signal is the vector who will undergo a change of base.
I hope that helps.
