Vector spaces | Linear algebra episode 1

#vectors #linearalgebra

What does the word "linear" mean, in terms of both algebra and geometry? In this video, we explore the rules for vector spaces. Along the way, we will talk about music, chaos theory, apples and oranges, quantum physics, and more.

The essence of linear algebra. The de facto starting point if you want to learn more.

Math the Beautiful. Detailed, very well explained. A great resource for beginners. Heavy focus on geometry rather than specific coordinates.

Over time, this is going to turn into a great reference on geometric algebra. But the introductory videos already give a very detailed account of how vector spaces work. The explanation is easy to follow, and starts from absolute zero.

The abstract definition of a vector space, followed by many concrete examples such as arrows, polynomials, and matrices.

This is a more formal and advanced introduction to vector spaces. Good if you already have a background.

Solving a sudoku by counting degrees of freedom.

0:00 Introduction
1:31 What does "linear" mean?
4:32 Non-linear systems
5:36 Applications (physics, AI, music)
8:51 The rules of vector algebra
11:50 Quantum superposition
14:00 Arrows are geometric vectors
15:48 How to subtract vectors
17:37 Linear combinations & spans
20:40 Degrees of freedom & constraints
23:08 Basis vectors
29:00 Coordinates are basis-dependent
32:54 The map of linear algebra

This video is published under a CC Attribution license

[halim_Ha]

I am sure your channel will reach millions believe me I just find it and hit subscribe

[alegian]

wow, Im glad you mentioned MTB! Professor Grinfield just answers all your questions the moment they arise!
he was the first youtuber that made me buy a book, his introduction to tensors :)

that being said I dare say there is a small gap at basic linear algebra, so I'm glad youre making this series

[DeathSugar]

Dark mode. Nice. But white on light blue is a bit hard to read - the two most lit one's at 10:31 addition and multiplication panels. Dimmed colors is more readable.

[AbElsaied]

what a beautiful video with unique flow of the motives behind the concepts and how to do it. Really thanks for your efforts.
Just wanted to ask are these animations by manim? Is there any simpler and faster way to do similar visuals?

[nanamacapagal]

Now I wish I knew how to bake a cheesecake...

[kanhaiyalalrajput]

Amazing! (I can't find any other great word to describe my feeling). I have watched many linear algebra YouTube videos, and many of them were great. But this video cleared some of my doubts that other videos failed to do so. Keep going on! 👍

[snuffybox]

Geometric Algebra has warped my brain... when you said we don't want to multiply vectors I almost cried

[martinsanchez-hwfi]

Awesome. Would love to know which tool you use for animations.
Edit: a little doubt: in 28:35, if the basis is not orthogonal, the decomposition would not be the projection, I think that statement in the video is wrong or what am I missing?

[linuxp]

Will you introduce basis (dual vector / covectors / linear forms / linear maps) as horizontal lists and coordinates (standard linear algebra vectors) as vertical lists?