1. Coordinate maps
consider a random vector a. If the values of every entry are explicitly given, one could naturally identify it as a vector under the orthonormal basis. However, it could also been understood as the coordinate vector, which represents a vector under another basis by a coordinate map:
2. Change of basis of a linear map
Now we consider a linear map (represented by a matrix) A. It could be interpreted as:
What if we want to change the basis of the objective vector space of f ? The following relation is obvious, if we want to find a matrix A’ representing the map after change of basis:
Where P is defined as a matrix that perform this basis change.
[Literature: Andre Lukas, Lecture note on Vectors and Matrices, University of Oxford]
3. The invariant map
Suppose we have a map that is invariant under any basis change, that is:
In other words, we would like to find an operator that commutes with any other operator on the same vector space V. Suppose now we have a vector x in V. We would like to find a non-trivial linear functional on x, so that we can define a linear map:
This is possible, for a functional in a n dimensional space can be express in to an (1 x n) matrix, so the left hand side can be expressed as (nx1)(1xn)(nx1) corresponding to v, f, and x respectively, and the former 2 combined and form a matrix. ( My reference mentioned ” Axiom of choice” with respect to finding non-trivial functional, and yet I have not understood it perfectly)
Then, the commutation relation implies:
Note that f(Tx), according to our pervious argument, should exist and independent on v. Thus, what we are doing consequently is that we have constructed linear maps P according to our need (that is, the vector v). In other words, we can assign every vector v in V a matrix P and they have to satisfy (6).
Then this inplies:
i.e. T is the scalar multiple of the identity.
[Literature: Robert Isreal, https://math.stackexchange.com/questions/27808/a-linear-operator-commuting-with-all-such-operators-is-a-scalar-multiple-of-the]