Normally, you need how system transforms n basis vectors to say how it transforms arbitrary vector. For instance, when your signal is presented in fourier basis, you need to know how system responds to every sine. But, I have noted that it is not true for the simplest standard basis. You just measure or compute a single Impulse Response (green's function?), for input <1000...> and you already know the responses for all other basis vectors: <0100...>, <0010...> and etc. You use it in convolution. How it is possible? Indeed, the delta-impulse, if you take its Fourier transform, is a combination of sines of constant amplitude. It is <111111...> in the Fourier basis. But, all sines are entangled here. So, instead of measuring response for every separate sine, we can measure response for all sines at once. Then, we can project the response to every separate sine and, thus, figure out the response per every separate sine. Is it right?