Using the discrete Fourier transform, we can benefit from representing digital signals in terms of frequencies rather than strength at different points in time. The transformation therefore takes a coordinate vector in the time basis and writes it in the frequency basis.
The information in digital signals, such as audio files or images, is encoded in the variation of the signal's strength over time. By finding the frequencies explaining these variations, the discrete Fourier transform lets us tweak the signal in various ways.
As an example, the technique can help us remove background noise from a sound recording to make it more crisp.
It can further be applied to compress a recording, by removing frequencies too high to be perceptible to the human ear.
A coordinate vector is essentially the same object as any other vector. However, the term lets us know that it is represented in a different form than we are used to.
Normally, a vector is written in the standard basis, where the first component of a vector in 2D represents its length in the x-direction and the second one the length in the y-direction.
We could instead choose two different directions to represent the same vector in another basis . By doing so we construct the coordinate vector .
Assume that we have a vector , expressed in the standard basis of , and we want to represent it in a different basis of .
For simplicity, let's consider the case , where:
In the basis , the same vector would be written as:
where and are the basis vectors in .
This change of basis can be achieved through multiplication with the transition matrix :
In fact, a transition matrix exist from any basis of the same subspace :
A coordinate vector is a vector whose coordinates are with respect to another basis. The name is used to emphasize that we are referring to a different basis than the standard basis. The notation usually looks like this:
Let be a vector in expressed in the standard basis, and let be a basis of . Then we note the coordinate vector of expressed in the basis as . We calculate it as follows:
A transition matrix is the connection between two different bases for the same subspace and whose multiplication results in a basis change of a vector. If you multiply the transition matrix by a coordinate vector with respect to the first basis, you get the coordinate vector with respect to the second basis. Formally, the following takes place:
Let the bases:
be bases for the subspace and . Then we note as the transition matrix from the basis to the basis . Then we have:
Furthermore, we define the elements of the transition matrix in the following way:
The transition matrix is also always invertible, and we have: