# Linear Algebra

### 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:

### Let’s get started

Let us start by talking about vectors in the plane:

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

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

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.

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.

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.

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.

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

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.

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

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.

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.

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.

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

Now, we are ready to define 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.

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.

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

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.

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

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

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