Here, you find my whole video series about Algebra in the correct order and I also help you with some text around the videos. If you want to test your knowledge, please use the quizzes, and consult the PDF version of the video if needed. In the case you have any questions about the topic, you can contact me or use the community discussion in Mattermost and ask anything. Now, without further ado, let’s start:
Part 1 - Introduction
Algebra is a video series I started for everyone who is interested in the basics of algebra as a mathematical field. It generalises a lot of ideas from my Start Learning Mathematics series and puts them into a more abstract setting. This is not only interesting from a pure mathematical point of view but we also find a lot of applications for these concepts. The power of describing them all under a single definition is what makes algebra so useful for a lot of different parts in mathematics.
With this you now know the topics that we will discuss in this series. Some important bullet points are groups, rings, fields and vector spaces. In order to describe these things, we need to generalise a lot from the constructions of number sets. Let’s immediately start with most basic structure.
Part 2 - Semigroups
In this video, we start with our first algebraic object, which is a binary operation. For this, we only need a set and inside we can combine two chosen elements. After this, we can make it more specific and define so-called semi-groups. They act as a preview to the more powerful object group, we will discuss later.
Part 3 - Identities and Inverses
After learning what a semigroup is, we can start to disguish special elements in it. The first one is a so-called neutral element.
Part 4 - Groups
Now we start with the definition of a group. We already know that they should contain an identity and inverses. However, it turns out that one can reduce the requirements without losing these properties. We present the nice and short proofs, which also show some standard calculation techniques in groups.
Content of the video:
00:00 Introduction
00:55 Semigroup with inverses
01:42 First definition of a group
03:16 Second definition of a group
05:00 Proof for equivalence of definitions
10:10 Credits
Summary of the course Algebra
- You can download the whole PDF here and the whole dark PDF.
- You can download the whole printable PDF here.