# Abstract Linear Algebra

#### 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.

###### Content of the video:

00:00 Introduction
00:36 Definition of a vector space
01:39 Examples
02:48 Function Spaces
08:01 Polynomial Space
10:00 Linear Subspace
11:55 Credits

#### 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.

###### Content of the video:

00:00 Introduction
00:36 Abstract vector space
01:38 Behaviour of the zero vector
03:00 Proof for the formulas
07:13 Linear subspaces = special vector spaces
10:00 Definition of linear subspace
14:07 Credits

#### 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.

###### Content of the video:

00:00 Introduction
00:36 Definition of polynomial spaces
02:27 Definition: linear combination
03:38 Definition: span of a subspace
04:44 Definition: generating set for a subspace
05:29 Definition: linear independent sets
06:36 Definition: basis of a subspace
07:57 Definition: dimension of a subspace
09:18 Examples for polynomial spaces
11:25 Example for infinite-dimensional vector space
11:44 Example for matrices
12:42 Credits

#### 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 $\ \Phi_{\mathcal{B}}$.

###### Content of the video:

00:00 Introduction
00:37 Assumptions
02:31 Definition: Coordinates with respect to a basis
03:40 Coordinate vector
03:58 Picture for the idea
06:44 Definition: basis isomorphism
08:32 Credits

#### 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.

###### Content of the video:

00:00 Introduction
00:33 Span in the function space
01:27 Is this a basis?
02:03 Checking for linear independence
05:40 Reformulating as matrix-vector multiplication
06:52 Uniquely solvable?
08:52 Basis isomorphism
11:16 Credits

#### 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.

#### Part 22 - Linear Maps

Here, we will discuss linear maps as we have done it for maps between $\mathbb{R}^n$ and $\mathbb{R}^m$ already. However, now we do it in the general setting for arbitrary general vector spaces. The definition looks the same since a linear map has to conserve the linear structure of the vector space.

#### Part 23 - Combinations of Linear Maps

A linear map conserves the linear structure of the vector space. It turns out that this is still the case if we combine linear maps in some special way. We discuss the addition and the composition of linear maps.

#### Part 24 - Homomorphisms and Isomorphisms

Let’s quickly talk about inverses of linear maps. As for the special linear maps $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, it turns out that inverses of linear maps are also linear. Usually one speaks of isomorphims in this case.

#### Part 25 - Matrix Representation for Linear Maps

A linear map between finite-dimensional vector spaces can completely be described by an ordinary matrix. One only has to fix two bases in domain and codomain.

#### Part 26 - Matrix Representations for Compositions

Matrix representations can help a lot if you want to combine some linear maps. In this case, one can just use the matrix product of the corresponding matrix representation if one chooses the same basis for the vector space in the middle. Let’s see how this looks in detail:

This course will get more videos in future! :)