Section 1 Introduction
The primary purpose of this document is to facilitate self-assessment of readiness for a first graduate level course on numerical linear algebra course. It was created for our "Advanced Linear Algebra: Foundations to Frontiers," (ALAFF) online resource, which is used by a Massive Open Online Course (MOOC) on edX by the same name.
Most of the questions use concrete problems similar to what a learner may have seen in a first undergraduate course on linear algebra. Being able to answer those questions is a necessary but not sufficient condition when starting a more advanced linear algebra course. To successfully complete a graduate level course on the subject, one must be able to master
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Abstraction.
This means that one must be able to take insights gained from manipulating concrete examples involving small matrices and vectors and extend these to the general case.
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Proofs.
It does not suffice to be able to execute an algorithm in order to compute a result. Abstraction allows one to state general principles (theorems, lemmas, corollaries), which must then be proved to hold via convincing arguments.
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Algorithms.
Numerical linear algebra is to a large degree about transforming theory into practical algorithms.
Once you have solved a given problem, it is important to spend time not just with the answer, but also with our suggested full solution. It is in the solution that you get an idea of how abstraction generalizes a concrete example, convincing arguments allow one to prove an assertion, and/or an insight can be translated into an algorithm.
In addition to poviding full solutions, we also discuss the relevance of the problem to more advanced linear algebra and/or to a course in numerical linear algebra. For those who find that they may a refresher, we point to where related material can be found in "Linear Algebra: Foundations to Frontiers," (LAFF) our online resource for a first undergraduate course in linear algebra
We hope this experience will help you get prepared!