This note goes over taking a derivative by use of Cartesian tensors.
Computation Cartesian Tensors PDF file
This note goes over the trace function on matrices.
Trace function on matrices PDF file.
More abstract features of linear algebra
Riesz Representation Theorem -- A simple version. Every linear mapping of a vector space (with an inner product) to its field of scalars can be represented by an inner product. The trick is to find this mystery vector in the inner product that appears 'out of the blue'.