This video is a blessing! I was googling about dual spaces for hours hoping to find so kind of illustrated explanation about the topic and I can say hands down that this video is the best explanation I found on dual spaces! It"s impressive how you managed to make it so clear and intuitive! I'm so glad I found your video. Many thanks!
Many thanks for the feedback and your kind words - I am flattered. Trying to find the balance between “hand wavy but intuitive” and “precise but abstract-nonsense“ is key, and I hope that I manage to find the balance. However, I am not sure whether I deserve your praise. Obviously there is no “optimal” method to explain math since there is no “optimal” way to explain anything really. I try to communicate math the way it lives in my head, which might not work for everyone. So it is definitely good to hear that someone feels the same way! ;-)
Outstanding explanation on the concept of dual vector space, thank you! Most videos focused on the mathematical nuts and bolts of the topic but not the reason "why" and the big picture.
Glad you enjoyed it! I however do not deserve your kind words, I simply try my best to avoid the bloody details: My (potentially flawed) approach is that an abstract formulation is awesome (e.g. you can write it down without ambiguity). It perfectly fits into a book, but maybe not into a TH-cam video. Some people might disagree with this approach.
@@VisualMath You are too modest! I like your approach and do hope you will produce more videos on Linear and Tensor Algebra topics...looking forward to it! In particular would appreciate if you could do some videos on contra and covariant vectors.
Oh, sorry if that wasn't clear: The capital K is the ground field (=where my scalars live). You can use K=rational numbers or K=real numbers, for example. So V*, formally, is the set of maps from V to scalars. In other words, the definition is general and works for all possible scalars. Anyway, thanks for the feedback. I am glad to hear that the explanation was helpful. I hope the video was enjoyable at the same time.
I do not need the "Sir" ;-), but way more importantly I am glad to hear that the video was helpful! Getting concept XYZ sometimes takes a while (same here!) and if anything I said was useful then this is awesome.
Yup, other sources on youtube that explain this topic are basically garbage compared to yours. My channel is with a graphics-programming bias, but I hope to carry the torch and pass down lucid explanations like yours in the future! Thank you! And you gained a subscriber today.
Thanks! I try give an intuition why one would define “Notion X” the way it is defined in most math classes. That is what usually works best for me. Its good to hear that this way of explaining maths works for other people as well. The other videos are not garbage, of course ;-), but rather take a different approach.
Dr. Tubbenhauer, Thanks for the great videos! what linear algebra books are your favorites? if you have to recommend 3 books to a beginner, which ones you would suggest?
Haha, I do not need any honorifics 😀 Anyway, here we go: **"Linear Algebra and Its Applications" by Gilbert Strang** **"Introduction to Linear Algebra" by Gilbert Strang** **"Linear Algebra Done Right" by Sheldon Axler** I hope that helps!
The pictures on the last slide were created using Mathematica, using either reference.wolfram.com/language/ref/Graphics.html or reference.wolfram.com/language/ref/Graphics3D.html depending whether the output is 2d or 3d. The used code is completely naive and nothing super exciting. Mathematica is just very good for illustrations.
Hmm, although that is not so far off, I would recommend to think of duality as its own thing. Its more an association than a formal mathematical operation in general if that makes sense.
This is a good question: Duality is a very general concept in mathematics that can be extended up to point where it gets ridiculously general. To answer your question, I have one video on duality in a categorical context th-cam.com/video/s8ZOoxlGXFY/w-d-xo.html but that might not be quite what you are looking for. (There are some nice diagrams, so maybe it is still helpful.) As one moves along in mathematics, one encounters various flavors of the vector space duality in this video, so I have no doubts that there will be videos in the future on duality.
Would it not be a lot simpler to draw a vector and then draw it’s dual partner. I have been searching for months to find any relation to what everybody talks about as dual basis, and that of geometry where you project a vector perpendicularly to get the dual basis
I am not sure what you are looking for. For example, in three space the perpendicular "vector" is a plane. So its somewhat better to use the pairing picture.
@@VisualMath Well, for those who find the topic quite difficult, maybe beginning with a vector in 2D space. Everyone understands that. Then speak on what relationship exists between a vector and a dual basis vector. From what I understand, a vector's components in 2D space can be represented by parallel projection (contravariant components and covariant basis vectors) OR by perpendicular projection (covariant components and contravariant basis vectors. All the videos I've seen except one, don't seem to mention these projections for skewed coordinate system dual basis vectors. Check out Daniel Fleish's "Vectors and Tensors for Students." What is the connection between the way you describe dual basis vectors and those derived geometerically from 1/|e1|cos theta? Thanks
This video is a blessing! I was googling about dual spaces for hours hoping to find so kind of illustrated explanation about the topic and I can say hands down that this video is the best explanation I found on dual spaces! It"s impressive how you managed to make it so clear and intuitive! I'm so glad I found your video. Many thanks!
Many thanks for the feedback and your kind words - I am flattered. Trying to find the balance between “hand wavy but intuitive” and “precise but abstract-nonsense“ is key, and I hope that I manage to find the balance.
However, I am not sure whether I deserve your praise. Obviously there is no “optimal” method to explain math since there is no “optimal” way to explain anything really. I try to communicate math the way it lives in my head, which might not work for everyone. So it is definitely good to hear that someone feels the same way! ;-)
Outstanding explanation on the concept of dual vector space, thank you! Most videos focused on the mathematical nuts and bolts of the topic but not the reason "why" and the big picture.
Glad you enjoyed it!
I however do not deserve your kind words, I simply try my best to avoid the bloody details: My (potentially flawed) approach is that an abstract formulation is awesome (e.g. you can write it down without ambiguity). It perfectly fits into a book, but maybe not into a TH-cam video. Some people might disagree with this approach.
@@VisualMath You are too modest! I like your approach and do hope you will produce more videos on Linear and Tensor Algebra topics...looking forward to it! In particular would appreciate if you could do some videos on contra and covariant vectors.
BTW I noticed a typo in your slide titled "Transpose Space". The product of row vector and column vector should be ac + bd and not ac + bc.
Good catch, awesome, thanks! You are right and I have put a warning in the description.
thank you so much! I'm glad I've found your chanel
I am glad that the video was helpful. Thank you for letting me know, any feedback is welcome.
This is a jaw dropping explanation ! Thank you so much! Just one question: What is the capital K at the 14.55 mark as part of the Definition of V* ?
Oh, sorry if that wasn't clear: The capital K is the ground field (=where my scalars live). You can use K=rational numbers or K=real numbers, for example. So V*, formally, is the set of maps from V to scalars. In other words, the definition is general and works for all possible scalars.
Anyway, thanks for the feedback. I am glad to hear that the explanation was helpful. I hope the video was enjoyable at the same time.
I was really struggling to grasp this concept. Thank you for your video sir. It genuinely helped me.😊😊
I do not need the "Sir" ;-), but way more importantly I am glad to hear that the video was helpful! Getting concept XYZ sometimes takes a while (same here!) and if anything I said was useful then this is awesome.
Yup, other sources on youtube that explain this topic are basically garbage compared to yours. My channel is with a graphics-programming bias, but I hope to carry the torch and pass down lucid explanations like yours in the future! Thank you! And you gained a subscriber today.
Thanks! I try give an intuition why one would define “Notion X” the way it is defined in most math classes. That is what usually works best for me. Its good to hear that this way of explaining maths works for other people as well.
The other videos are not garbage, of course ;-), but rather take a different approach.
Dr. Tubbenhauer, Thanks for the great videos! what linear algebra books are your favorites? if you have to recommend 3 books to a beginner, which ones you would suggest?
Haha, I do not need any honorifics 😀
Anyway, here we go:
**"Linear Algebra and Its Applications" by Gilbert Strang**
**"Introduction to Linear Algebra" by Gilbert Strang**
**"Linear Algebra Done Right" by Sheldon Axler**
I hope that helps!
@@VisualMath Thanks!
@@sakcee Welcome ☺
Btw is there some software you used to make the diagrams at the end? Or were they downloaded from the internet?
The pictures on the last slide were created using Mathematica, using either
reference.wolfram.com/language/ref/Graphics.html
or
reference.wolfram.com/language/ref/Graphics3D.html
depending whether the output is 2d or 3d. The used code is completely naive and nothing super exciting. Mathematica is just very good for illustrations.
Is duality a kind of isomorphism?
Hmm, although that is not so far off, I would recommend to think of duality as its own thing. Its more an association than a formal mathematical operation in general if that makes sense.
Do you have any more videos that touch on duality?
This is a good question: Duality is a very general concept in mathematics that can be extended up to point where it gets ridiculously general.
To answer your question, I have one video on duality in a categorical context
th-cam.com/video/s8ZOoxlGXFY/w-d-xo.html
but that might not be quite what you are looking for. (There are some nice diagrams, so maybe it is still helpful.)
As one moves along in mathematics, one encounters various flavors of the vector space duality in this video, so I have no doubts that there will be videos in the future on duality.
15:37 it should be f*(\phi) = \phi compose f . The star is missing
Thanks for catching the typo: yes that is correct 👍
2:30 needs prospective geo
Not sure what you mean. Can you elaborate?
Would it not be a lot simpler to draw a vector and then draw it’s dual partner.
I have been searching for months to find any relation to what everybody talks about as dual basis, and that of geometry where you project a vector perpendicularly to get the dual basis
I am not sure what you are looking for. For example, in three space the perpendicular "vector" is a plane. So its somewhat better to use the pairing picture.
@@VisualMath Well, for those who find the topic quite difficult, maybe beginning with a vector in 2D space. Everyone understands that. Then speak on what relationship exists between a vector and a dual basis vector.
From what I understand, a vector's components in 2D space can be represented by parallel projection (contravariant components and covariant basis vectors) OR by perpendicular projection (covariant components and contravariant basis vectors. All the videos I've seen except one, don't seem to mention these projections for skewed coordinate system dual basis vectors. Check out Daniel Fleish's "Vectors and Tensors for Students." What is the connection between the way you describe dual basis vectors and those derived geometerically from 1/|e1|cos theta?
Thanks