Trace of a linear operator

[strugglinginmat]

I understand the definition of trace and linear operator individually but I don't seem to understand as to what does it mean by trace of a linear operator on a finite dimensional linear space.
What I have found out is that trace of a linear operator on a finite dimensional linear space is the trace of any matrix which represents the operator relative to an ordered basis of the space. I am confused as why is this definition well defined.
If T:V->V is the linear operator defined on V by T(A)=BA for all A in V and B is a fixed matrix. How do I represent T relative to standard ordered basis for V where V is the linear space of all 2X2 real matrices.

 

[CompuChip]

As for the first part: you can prove it as follows: Let L be a linear operator and A its matrix representation w.r.t. some chosen basis.
- Check that the representation w.r.t. any other basis can be written as $$D A D^{-1}$$, where D is an invertible ("change of basis") matrix.
- Now check the cyclic property for the trace (e.g. by writing out in components) $$\mathrm{Tr}(A B C) = \mathrm{Tr}(B C A) = \mathrm{Tr}(C A B)$$
- Combine them, you'll see that the trace is the same in any basis. So it's well-defined.

Now in general, you can write out the matrix of a linear transformation by finding out how it acts on the basis vectors. For example, suppose A mirrors the plane in the origin. Take the standard basis i = (1, 0), j = (0, 1). Now Ai = (-1, 0) and Aj = (0, -1). Putting these as columns of A gives $$A =\left(\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right)$$. Now check for yourself, that this does indeed produce the correct result for any vector (hint: write it out in components w.r.t. to the basis {i, j}).

Hope that gets you started. I left the details out on purpose, if you get stuck anywhere just ask :)

 

[tacman]

The trace of a linear operator on a finite dimensional linear space is a numerical value that represents the sum of the diagonal elements of the matrix representation of the operator. In other words, it is the sum of the eigenvalues of the operator.

In the example given, the linear operator T is defined as T(A)=BA, where B is a fixed matrix. To represent T relative to the standard ordered basis for V, we need to determine the matrix representation of T with respect to this basis. This can be done by finding the matrix representation of B and then multiplying it by the matrix representation of A. The resulting matrix will be the matrix representation of T relative to the standard ordered basis.

The reason why this definition is well-defined is that the trace of a linear operator is independent of the choice of basis. This means that no matter which basis we choose to represent the operator, the trace will remain the same. This is because the trace is a property of the operator itself, not of its matrix representation. Therefore, the trace of a linear operator is a unique and well-defined value.

In summary, the trace of a linear operator on a finite dimensional linear space is a numerical value that represents the sum of the diagonal elements of the matrix representation of the operator. It is independent of the choice of basis and is a unique and well-defined value.