# Abstract Linear Algebra

#### Part 1 - Abstract Linear Algebra

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.