Applications of the rank plus
nullity theorem.
A set of problems to vindicate my (possibly eccentric) opinion
that the rank plus nullity theorem is one of the great unsung
existence theorems of elementary mathematics. This includes examples
of applications of the result in algebra, ordinary and partial
differential equations and algebraic geometry.
A gentle introduction to rings and
modules. The proof of the normal forms for linear maps and
matrices was done via the structure of finitely generated
modules over a Euclidean domain. This is a introduction to rings up
to a proof of fundamental theorem of arithmetic (i.e. unique
factorization into primes) in a Euclidean domain. Many of the proofs
are exercises with generous hints.
Various problems.
A collect of problems not in the book that were assigned during
the term.
Qualifying exam problems.
A collect of the linear algebra problems from the Admissions to
Candidacy Exams for the 1984--1995.
The characteristic polynomial of a
product. A proof that for square matrices A and B
that AB and BA have the same characteristic
polynomial and thus the same set of eigenvalues. I have had to work
this problem a couple of times a year for students studying for the
qualifying exam and so TeXed it up to save time in the future.
