Coordinate (lame) version

Given a square matrix the trace is the sum of the diagonal entries, the trace of is

In this form its a bit hard to see why it is important has has some interesting properties.

Coordinate free version

For finite dimensional vector spaces and , there is a canonical isomorphism using the tensor product

So we can use this idea on the set of linear maps to make a new linear map .

Thus, the trace of a linear map is

as above.

How do these two definitions match?

Given a basis for , then

Then looking at each then we can see that

So, all together, (combining coefficients) this is equivalent to

where is the dual basis.

So then, using the definition of the trace, we get

since for . This coincides with the trace formula for a matrix form of a linear map.

@brocker2003 - Chapter 2 section 3