Here, you find my whole video series about Abstract Linear Algebra in the correct order and you also find my book that you can download for free. This series extends the original Linear Algebra course. On this site, I also want to 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 - Abstract Linear Algebra
Let’s start the series by defining a general vector space. From now on, we will always remember that we need 8 rules to check if something is a vector space.
Content of the video:
00:00 Introduction
00:28 Prerequisites
01:28 Overview
03:40 Definition of a vector space
05:56 Operations on vector spaces
08:10 8 Rules on these operations
Part 2 - Examples of Abstract Vector Spaces
In this video, we will look back at the set of matrices and show again that they form a vector space. In addition, we will also look at more abstract examples, like function spaces and polynomial spaces. There we will already see that a lot of notions from the Linear Algebra course can be reused.
Part 3 - Linear Subspaces
In this third part, we are finally ready to generalize the notion of subspaces. We will see that the polynomial space is a subspace contained in the general function space.
Part 4 - Basis, Linear Independence, Generating Sets
Now we are ready to generalize the notion of a basis for abstract vector spaces. Essentially, the definitions look exactly the same as for $ \mathbb{R}^n $ and $ \mathbb{C}^n $. However, it’s good to formulate them again and point out the small differences in this generalization.
Part 5 - Coordinates and Basis Isomorphism
After fixing a basis in an abstract vector space, one can use that to translate this space to the very concrete space $ \mathbb{F}^n $. This translation is called the basis isomorphism, written as $ \ \Phi_{\mathcal{B}} $.
Part 6 - Example of Basis Isomorphism
This video explains the basis isomorphism with some examples. We see that it is a very natural construct that helps to analyze abstract vector spaces, as long as they are finite-dimensional. In particular, we show how we can prove that vectors from a function space like $ \mathcal{P}(\mathbb{R}) $ are linearly independent.
Part 7 - Change of Basis
The next videos will cover the subject of changing a basis in a vector space. This means that we have to combine twwo basis isomorphisms. This change of basis is very important because often we want to choose a suitable basis for solving a given problem. Hence, we need to know how switch bases in an effective way.
Part 8 - Transformation Matrix