*Here, you find my whole video series about Linear Algebra in the correct order and you also find my book that you can download for free. 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. When you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:*

#### Part 1 - Introduction

**Linear Algebra** is a video series I started for everyone who is interested in calculating with vectors and understanding the abstract ideas of vector spaces and linear maps. The course is based on my book Linear Algebra in a Nutshell. We will start with the basics and slowly will climb to the peak of the mountains of Linear Algebra. Of course, this is not an easy task and it will be a hiking tour that we will do together. The only knowledge you need to bring with you is what you can learn in my Start Learning Mathematics series. However, this is what I explain in the first video.

With this you now know already some important notions of linear algebra like **vector spaces**, **linear maps**, and **matrices**. Now, in the next video let us define the first vector space for this course. Some more explanation, you can find in my book:

#### Part 2 - Vectors in $ \mathbb{R}^2 $

Let us start by talking about **vectors** in the plane:

#### Part 3 - Linear Combinations and Inner Products in $ \mathbb{R}^2 $

Now we talk about **linear combinations**, the standard **inner product** and the **norm**:

#### Part 4 - Lines in $ \mathbb{R}^2 $

With this, we are now able to define **lines** in the plane:

#### Part 5 - Vector Space $ \mathbb{R}^n $

Now we are ready to go more abstract. Let’s define a general **vector space** by listing all the properties such an object should satisfy. We can visualise this with the most important example.

#### Part 6 - Linear Subspaces

In the next video, we will discuss a very important concept: **linear subspaces**. Usually we just call them **subspaces**. They can be characterised with three properties.

#### Part 7 - Examples for Subspaces

I think that it will be very helpful to look closely at some **examples** for subspaces. Therefore, the next video will be about explicit calculations.

#### Part 8 - Linear Span

The next concept we discuss is about the so-called **span**. Other names one uses for this are **linear hull** or **linear span**. It simply describes the smallest subspace one can form with a given set of vectors.

#### Part 9 - Inner Product and Norm

As for the vector space $ \mathbb{R}^2 $, we can define the standard **inner product** and the **Euclidean norm** in the vector space $ \mathbb{R}^n $.

#### Part 10 - Cross Product

In the next part, we will look at an important product that only exists in the vector space $ \mathbb{R}^3 $: the so-called **cross product**.

#### Part 11 - Matrices

When we want to solve systems of linear equations, it’s helpful to introduce so-called **matrices**:

#### Part 12 - Systems of Linear Equations

After introducing matrices, we can now see why they are so useful. They can be used to describe **systems of linear equations** in a compact form.

#### Part 13 - Special Matrices

In the next video, we go back to matrices. We will discuss some important **names for matrices**, like square matrices, upper triangular matrices, and symmetric matrices.

#### Part 14 - Column Picture of the Matrix-Vector Product

Let’s continue talking about the important matrix-vector multiplication we introduced while explaining systems of linear equations. In the next video, we discuss the so-called **column picture** of the matrix-vector product.

#### Part 15 - Row Picture

Similarly, we can look at the rows of the matrix, which leads us to the **row picture** of the matrix-vector multiplication.

#### Part 16 - Matrix Product

Now, we are ready to define the **matrix product**.

#### Part 17 - Properties of the Matrix Product

After defining the matrix product, we can go into the details and check which **properties** for this new operation hold and which don’t.

#### Part 18 - Linear Maps (Definition)

Let’s go more abstract again: we will consider so-called **linear maps**. They are defined in the sense that these maps conserve the linear structure of vector spaces.

#### Part 19 - Matrices induce linear maps

By knowing what a linear map is, we can look at some important examples. It turns out that all **matrices induce linear maps**.

#### Part 20 - Linear maps induce matrices

The converse of the statement of the previous video is also true. All **linear maps induce matrices**. This is an important fact because it means that an abstract linear map can be represented by a table of numbers.

#### Part 21 - Examples of Linear Maps

Linear maps preserve the linear structure. This means that linear subspaces are sent to linear subspaces. Let’s consider some **examples**.

#### Part 22 - Linear Independence (Definition)

In the following video, we consider a new abstract notion: **linear dependence** and **linear independence**. We first explain the definition.

#### Part 23 - Linear Independence (Examples)

Now let’s consider **examples** of linearly independent families of vectors.

#### Part 24 - Basis of a subspace

Let’s discuss the concept of a **basis**.

#### Part 25 - Coordinates with respect to a Basis

Next, let’s talk how we calculate with bases. We define **coordinates** of vectors with respect to a chosen basis. Depending on the problem you want to solve, different bases might be helpful such that the coordinates you calculate with are simpler.

#### Part 26 - Steinitz Exchange Lemma

The next part will be more technical and about the so-called **Steinitz Exchange Lemma**. This will be used in some proofs later.

#### Part 27 - Dimension of a Subspace

After this technical proof, we are now able to define the concept of **dimension**. This is a natural number that describes the number of degrees of freedom in a subspace.

#### Part 28 - Conservation of Dimension

The dimension has a nice property that is **conserved** is some sense for linear maps.

#### Part 29 - Identity and Inverses

In the next videos, let us discuss some more concrete objects again. First, we look at matrices and define a special one: the **identity matrix**.

#### Part 30 - Injectivity, Surjectivity for Square Matrices

We recall the important notions for maps: **injectivity** and **surjectivity**. This concepts also hold for linear map and, therefore, can be transferred to matrices as well. Especially for square matrices, we find a very nice connection:

#### Part 31 - Inverses of Linear Maps are Linear

Let us quickly prove the important fact that, for bijective linear maps, the inverses are always also linear.

#### Part 32 - Transposition for Matrices

In the next video, we define a matrix operation: the **transpose**.

#### Part 33 - Transpose and Inner Product

The tranpose of a matrix could also be defined by using the standard inner product. This is an important relation that explains why the transpose is such a useful object.

#### Part 34 - Range and Kernel of a Matrix

The following two definitions are very important for the rest of the course: the **range** of a matrix and the **kernel** of a matrix. Both are defined as sets in a vector space and it turns out that they are actually subspaces.

#### Part 35 - Rank-Nullity Theorem

Let’s immediately use the definitions from above in order to formulate a key property of linear maps and matrices: the **rank-nullity theorem**. In addition, we will also be able to prove this fact.

#### Part 36 - Solving Systems of Linear Equations (Introduction)

Now we go back to our motivation while doing linear algebra: we want to **solve systems of linear equations**. This video is like an introduction into this huge topic. After getting the rough idea how the solving process should work, we will go into more details with later videos.

#### Part 37 - Row Operations

What we need to solve systems of linear equations are so-called **row operations**. These can be described by using invertible matrices that are multiplied from the left-hand side.

#### Part 38 - Set of Solutions

Here, we will define the set of solutions for a system of linear equations. It’s denoted by $ \mathcal{S} $.

#### Part 39 - Gaussian Elimination

This is one of the most important topics in this course. It is not theoretically complicated but the applications are everywhere. The Gaussian elimination is always needed when you want to solve a system of linear equations in an algorithmic way.

#### Part 40 - Row Echelon Form

Now, we are ready to define the important end result for the Gaussian elimination. This generalises a triangular form we have seen in former videos. It’s called **row echelon form** because the structure is given like a staircase.

#### Part 41 - Solvability of a System

Let’s discuss our result more abstractly now. We can always transform a system of linear equations into row echelon form. What does this say about the *solvability* of the system. It turn out that we can nicely formulate equivalent statements there.

#### Part 42 - Uniqueness of Solutions

The upcoming video will show in which cases we have a unique solution for a system of linear equations. This will be also the last video about the Gaussian elimination for now.

#### Part 43 - Determinant (Overview)

In the next videos, we will talk a lot about so-called **determinants**. First we will motivate them:

#### Part 44 - Determinant in 2 Dimensions

In this video, we will see why a determinant makes sense for solving systems of linear equations and how we can calculate this determinant in two dimensions.

#### Part 45 - Determinant is a Volume Measure

Here, we introduce the notion of a volume measure in arbitrary dimensions. Please note that in 2 dimension, this coincides with the common area function we already discusses in the last video. Hence, we already now which rules a general volume function should fulfil. It turns out that these rules already determine the volume measure.

#### Part 46 - Leibniz Formula for Determinants

From the last video we can conclude a nice formula for the volume measure and, therefore, also for the determinant.