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:
Part 13 - Orthogonality
After discussing general inner products, we know that they can describe geometry in general vector spaces. In particular, the notion of orthogonal vectors make sense. So even in abstract vector spaces, we can say that the vector $ x $ is perpendicular to $ y $ and we write it as $ x \perp y$.
Part 14 - Orthogonal Projection Onto Line
Here we start with an important topic: orthogonal projections. It’s a very visual concept if you imagine an object that produces a shadow because the sun is above it. Indeed, this gives exactly the correct picture if you represent a vector by an arrow and introduce a right-angle for this projection. The nice thing is that this visualization also works in our abstract vector spaces. Let’s start this disussion in a one-dimensional case.
Part 15 - Orthogonal Projection Onto Subspace
We already know the orthogonal projection onto a one-dimensional line. Now, we should be able to generalize this to a finite-dimensional subspace of our vector space. Indeed, the visualization looks very similar because we still search for a decomposition $ \mathbb{x} = \mathbb{p} + \mathbb{n} $ where $ \mathbb{p} $ lies in the subspace and $ \mathbb{n} $ is orthogonal to it.
Part 16 - Gramian Matrix
To calculate the orthogonal projection from the last part, we need to solve a system of linear equations. This one can be represented by the so-called Gramian matrix.
Part 17 - Approximation Formula
After all the calculations for orthogonal projections, we can give another motivation why these projections are so important, especially in the abstract settings. The approximation formula shows that the orthogonal projection minimizes the distance between the point and the subspace, a property that can be useful whenever such a minimizer is needed.
Part 18 - Orthonormal Basis
For calculating the orthogonal projection, we had to solve a system of linear equations given by the Gramian matrix. The system would be already completely solved if the matrix is in diagonal form.This leads to the notion of an orthogonal basis and of an orthonormal basis, usually abbreviated by (ONB).
Part 19 - Fourier Coefficients
The name Fourier expansion and Fourier coefficients are often used in a special context, discussed in the video series Fourier Transform. However, they also make sense in the abstract description of an ONB in a general vector space.
Part 20 - Gram-Schmidt Orthonormalization
In the following, we will explain how we take any basis in a finite-dimensional vector space with inner product and transform it into an orthonormal basis. This is known as the Gram-Schmidt process. It is useful when you need an ONB to make your calculations simpler. We will seed that the algorithm is not complicated at all since it essentially just uses the orthogonal projections we already know.
Part 21 - Example for Gram-Schmidt Process
After just explaining the algorithm in the last video, we can put some life to it. It’s a good exercise to do some orthonormalization in the vector space $ \mathbb{R}^n $ to see the algorithm in work. However, here we present a more abstract example: we will consider polynomials. In fact, these procedure will lead to the so-called Legendre polynomials.
This course will get more videos in future! :)
Connections to other courses
Summary of the course Abstract Linear Algebra
- You can download the whole PDF here and the whole dark PDF.
- You can download the whole printable PDF here.
- Test your knowledge in a full quiz.