The first set of lectures (Week 0, since as we know python list are indexed starting with 0!) has little to do with actual linear algebra, but instead gives you quick intros to two central concepts: functions and fields:
The first lectures (The Function) review some basic notation, terminology and concepts that you’ll be using throughout the course, including notation for sets and functions. The highlight of this part is the concept of invertible function.
After that, we’ll look at The Field. A field is a collection of values with a plus operation and a times operation. You will already be familiar with the field of real numbers but perhaps not with the field of complex numbers or the field GF(2), which consist just of zero and one. We will discuss these fields and give examples of applications.
Functions you are hopefully already familiar with, from precalculus, calculus, etc. Professor Klein’s lectures go over the concepts and terminology fairly quickly, so you may need to pause as you watch, and perhaps consult some other material. If you have your textbook from when you took precalc or calc, take a look at that section(s) in there on functions. Or you can take a look at:
- The Wikipedia entry for “Function (mathematics)“
- Khan Academy’s set of videos on Functions
- A math prof at Lamar Univ has a bunch of online math notes and tutorials. Take a look at Pauls Online Notes: The Definition of a Function
Fields you may be less familiar with. Actually, as noted above, you’ve worked with the canonical example of a field in your previous math classes–the real numbers. But to illustrate the general concept, Professor Klein introduces two other fields in his lectures: the complex numbers and the Galois Field with two elements, abbreviated GF(2).
The important feature of fields is that they include both an addition and multiplication–these are the two operations needed to carry out linear algebra! Soon we’ll go over vectors and matrices “over” any given field (meaning the entries in the vector or matrix are taken from the given field).
(Btw, GF(2) is named for the famous algebraist Evariste Galois–there’s a whole branch of abstract algebra called Galois theory, which he developed during his short life. In fact, finite fields in general are called Galois fields in his honor.)
For more on the complex numbers and GF(2), see:
- Khan Academy’s section on complex numbers
- Wolfram MathWorld’s entries on complex numbers and the complex plane
- this set of lecture notes on GF(2)
