Abstract Linear Algebra

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 - Vector Space

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.


YouTube YouTube Dark PDF Quiz


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.


YouTube YouTube Dark PDF Quiz


 

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.


YouTube YouTube Dark PDF Quiz


 

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.


YouTube YouTube Dark PDF Quiz


 

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}} $.


YouTube YouTube Dark PDF Quiz


 

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.


YouTube YouTube Dark PDF Quiz


 

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.


YouTube YouTube Dark PDF Quiz


 

Part 8 - Transformation Matrix

Now we can put the change-of-basis map in the form of a matrix. This change-of-basis matrix has some different names, like transformation matrix or transition matrix. However, it simply describes what happens to the basis vectors like what we learnt in the original Linear Algebra course.


YouTube YouTube Dark PDF Quiz


 

Part 9 - Example for Change of Basis

After all this theoretical talk, we finally want to look at concrete example. Indeed, we can take the vector space $ \mathbb{R}^2 $ and do a change of basis there. It turns out that this is a general calculation scheme one can also use in higher dimensions.


YouTube YouTube Dark PDF Quiz


Content of the video:

00:00 Introduction
00:39 Change-of-basis matrix
01:58 Example - explanation
02:50 Example in R²
05:10 First transition matrix
05:54 Second transition matrix
06:39 Composition of change-of-basis
07:51 Gaussian elimination for product
11:41 Credits


 

Part 10 - Inner Products

Let’s go to an easier topic again. We already know from the Linear Algebra course that measuring lengths and angles can be generalized. Now, we can also describe inner products for more abstract vector spaces, like the polynomial space.


Early Access PDF Quiz


 

Part 11 - Positive Definite Matrices

This is very concrete topic about matrices. However, since positive matrices play a crucial role for defining inner product, we discuss them now in more detail.


Early Access PDF Quiz


 

Part 12 - Cauchy-Schwarz Inequality

You can remember that inner product are used to give a geometry to the the vector space. This means that it is possible to measure angles between vectors and lengths of vectors. In particular $ | x | = \sqrt{\langle x, x \rangle} $ defines a so-called norm. The relation to the inner product and this iduced norm is stated in the famous Cauchy-Schwarz inequality, which will prove now:


Early Access PDF Quiz


 


Summary of the course Abstract Linear Algebra