It is known that the Fourier transform, continuous and discrete, can be interpreted as a change of basis. The basis is the one formed by complex exponentials.
Each basis vector is a complex exponential of different (integer or real for the continuous FT) frequency, yes?
Can we say that these basis vectors (complex exponentials) for that basis of a Complex vector space ?
Can you confirm/correct/formalize my statements? (The difficulty here is the notion of complex vector space)
*(For the DFT (1D) I like to see it as a matrix multiplication (2D DFT matrix with basis vectors in each row times 1D vector in time domain).
More concretely, for the DFT, if we have 8 vectors, each containing 8 points of complex exponentials of different frequencies (all orthogonal to each other), i.e., in Matlab:
B=dftmtx(8)
can we safely say that those 8 vectors form a basis of $\mathbb{C}^8$ ?!