Intro
When musicians hear a chord, they can often pick out the individual notes that make up that chord.
The musicians are actually doing something very mathematical: they are performing a Fourier transform.
What the Fourier transform does is to pick out all the individual notes that make up a sound wave.
The Fourier transform is an example of a Linear transformation.
Concept
When multiplying a vector by a matrix, we change its coordinates to get another vector back, which we can interpret as rotated, stretched or squeezed in some way.
Alternatively, we can also view this change of a vectors coordinates as a way of representing the vector in a new form, or basis.
It is sometimes beneficial to imagine that our matrix multiplication do both. In other words, we change the vector's direction and length, and simultaneously represents it in a new way.
The reason is that there is a simple relationship between the linear transformation and the change of basis of the vector that we can exploit.
Math
Let's say we have a vector in the standard basis of the subspace and we want to both apply a linear transformation , and represent it in a different basis of the same subspace.
The following scheme represent the relationship between a linear transformation and a change of basis tells us that we now have two alternatives:
Multiply first by the transition matrix to represent in the basis , and then by , the standard matrix of the linear transformation with respect to .
Start by multiplying with the standard matrix , before changing the basis of this product vector to through multiplying it by .
The fact that:
This let's us choose whichever method is more convenient for a particular problem.
Linear transformations and bases
There are problems in linear algebra that mix linear transformations and basis changes. These are usually perceived as the most difficult to solve. They all involve matrix multiplication, which can be described as:
which can be confusing, but in short, the basis switch is a linear transformation! Let us, on the one hand, consider as a standard matrix, and on the other hand, as a transition matrix.
Standard matrix
Let be a linear transformation. Its standard matrix is then expressed with respect to the standard basis as:
Transition matrix
Let:
be bases for . Then we have that the transition matrix from the basis to is expressed as:
Standard matrix with respect to another basis
As we have discussed, a matrix can refer to both a standard matrix for a linear transformation and a transition matrix for a basis change. If we want to produce the standard matrix for the linear transformation with respect to the basis , we are free to calculate the following (which follows analogously from the two expressions above):
However, the calculation above takes quite a lot of computing power (both for computers and for humans), and therefore it is sometimes appropriate to use the following scheme:
The scheme shows the relationship between the matrices, (standard matrix), (transition matrix) and (the standard matrix with respect to the basis ), and the vectors, , , and . The scheme is read as follows: suppose we start from vector and want to express the transformation and the coordinate vector . We achieve this by means of matrix multiplication by and , respectively:
If we look further at the scheme, we see two expressions taking the form of based on the vector , via a half turn clockwise and counterclockwise:
This implies that the matrix multiplication must be equal to . The following equations are thus equivalent and everyone can choose their favorite:
