Matrices
The purpose of this post is to explain matrix multiplication, the trace and the transpose in terms of duality and tensor products.
Let 
be a finite dimensional vector space, and let 
be its dual. We are interested in the two vector spaces 
and 
. On account of the duality we have a map:
with the property that for any basis 
of there exists a unique dual basis 
of such that:
We also have a map:
As vector spaces and are naturally isomorphic:
Now, given a matrix:
We can either think of it as a vector in
and write it as:
or we can think of it as a vector in
and write it as:
Here we are making use of the physicists summation convention. In the first interpretation the usual trace operator is given by 
while in the second it is given by 
.
It is possible to define a multiplication on via:
Similarly, it is also possible to define a multiplication on via:
Making use of the notation 
, we have:
Similarly, making use of the notation 
, we have:
From this we observe the following:
In conclusion, a matrix may be thought of as either as an element of or as an element from 
. It is a question of interpretation.
The spaces 
and 
are isomorphic as vector spaces but anti-isomorphic as rings.