The discrete Fourier transform, as one of many possible orthogonal transforms (e.g., discrete cosine transform, etc.), can be considered as a rotation of the standard basis vectors. A signal as a N-D vector is represented implicitly by the standard basis. It can be equivalently represented by any other basis, some rotated version of the standard basis. This idea can be illustrated by the following example of an N=2 dimensional vector space.
A 2-D vector space is spanned by two standard basis vectors:
can be considered as a discrete signal of samples:
In fact, this rotation is the discrete Fourier transform of signal of samples:
The N-point Discrete Fourier transform is essentially a rotation of the standard basis vectors to a set of new basis vectors representing different frequencies. More generally, each of the infinitely many possible rotations of the standard basis vectors corresponds to a certain orthogonal transform, and the Fourier transform is only one of these transforms. Also as an orthogonal transform is simply a rotation of the basis of the vector space, it does not change the norm (length) of a vector, i.e., the energy contained in a signal is conserved before and after the transform, this is the Parseval's identity.