*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.

###### 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

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.

#### 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.

###### 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.

#### 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.

#### 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:

#### Summary of the course Abstract Linear Algebra

- You can download the whole PDF here.
- You can download the whole printable PDF here.
- Test your knowledge in a full quiz.