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Some resources relating to preparation for a graduate numerical linear algebra class.

Numerical linear algebra, and more broadly numerical analysis, is equal parts a continuation of real analysis and practical implementation. It is hard for many since it is a blend of theorems, linear algebra, analysis, as well as problem solving and coding. Understanding and appreciating the nuances via numerical experimentation is essential.

Assuming a student is going to be taking a course based on Numerical Linear Algebra by Trefethen and Bau (for example), here are some reasonable representative resources one could use to prepare:

• From the linear algebra side, Axler's Linear Algebra Done Right; link; it's likely you can obtain digital copies through your university library or similar. The following are the minimal pieces necessary:
  • Chapter 1B, 1C,
  • Chapter 2,
  • Chapter 3A-3C,
  • Chapter 5,
  • Chapter 6A, 6B,
  • 7B, 7D* (for SVD)
• From the analysis side, Introduction to Functional Analysis by Daners (up to and including the chapter on Hilbert spaces)
• If you want a single book, or prefer inexpensive physical books, I'm a fan of books by Georgi Shilov;
  • "Linear Algebra" by Shilov is similar but doesn't get into the analysis as much.
  • "Elementary Functional Analysis" by Shilov is the most abstract (after a hundred pages of theory, the following section applies this to find the "usual" solutions linear n-th order differential equations)
  • "An introduction to the theory of linear spaces" by Shilov is in the Goldilocks zone, covering a little bit of everything. A digital copy exists on archive.org if you make a free account; link (current as of 3 August 2023); otherwise you might have to interlibrary loan or buy if you want a physical copy.
• A very brief intro to the essentials of asymptotics by Senning: link
• Programming: Hard to point at a specific resource. But the necessary skills are:
  • for loops, while loops, etc.
  • knowing how to make functions,
  • run parameter sweeps,
  • make loglog plots to study convergence,
  • make plots of matrices (the "spy" or "matshow" command in matlab or pyplot),
  • understanding of data types and how to work with vectors/matrices/arrays.
  • A thing to get your feet wet in a lot of this is to produce plots studying different methods for rootfinding (studying how to solve f(x)=0 for arbitrary function f) — Newton's method, bisection, secant, false position method.
• Yes, these topics have overlap.

                                                             
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  ("It's all linear algebra?" "Always has been.")