I have some troubles with understanding time domain, not on the intuitive level, but in math terms.
For example I have a vector signal $$ x = [x_0,x_1,x_2,...,x_{N-1}]$$ I understand that generally when we describe our signal in the form of a vector, we just put our information in a convenient form for further manipulation.
i can change the basis to the frequency domain via DFT analysis, and then I can get back (to time domain) with DFT synthesis. DFT synthesis just multiply my complex amplitude on complex sinudoids with different weights $w$. And it's perfectly correct on physics level like we just sum some periodic functions with different amplitudes.
But when we are talking about change of basis, we have in mind:
$$\alpha_k = \sum_{k=0}^{N-1} \beta_n \langle w^{(k)},w^{(n)} \rangle$$
where: $w^{(k)}$ - the first basis ; $w^{(n)}$ - the second basis ; $\alpha_k$ - expansion coefficients in second basis; $\beta_n$ - expansion coefficients in first basis;
where:$$x_n = \sum_{n=0}^{N-1}\beta_nw_n$$ - this is our information vector in time domain; and basis for frequency domain is $$w^{(k)} = w_k[n] = e^{\frac{2\pi i}{N}kn}$$
and my question is: do I understand right that time domain basis is a canonical basis for N-dimensional linear space like that: $$\hat{e}^{(k)} = \delta[n-k]$$