Quick request: maybe you could do a side video reconciling the notion that geometric vectors all proceed from one arbitrarily chosen point (on the one hand) with the tip-to-tail rule (on the other), which requires that they be all over the place? It looks to me that this would require a not entirely trivial construction involving equivalence classes of lines or rays, plus whatever else it takes to define 'sense' in the directional sense. One point of interest is that this would illustrate what it takes to advance from intuition to consistency --- a central issue in almost all kinds of thought.
Good comment! To be honest, I don't see much of an issue. Tip-to-tail is just a set of instructions for constructing the sum and it doesn't apply that vectors can float around the space. But even if you do allow vectors to float around the space - yes, you can introduce an equivalence class where two vectors are equivalent if they are related by a parallel shift. Then you're back to having a simple vector space. The only reason I avoid considering that is that I err on the side of less.
Professor, did you study in France or in a French speaking country? Or have some French colleagues? Because "sens" and "direction" is a distinction I never hear in English courses! Only in France! Superb channel. Long time fan here
No, I studied in Russia where we only had one word "direction". I discovered the word "sense" in a book I recently read, but I don't remember which. If I rediscover it, I'm come back here and share it.
You can’t introduce dU without complex numbers. Similarly you can’t claim scalar vector multiplication scales the magnitude but then glibly introduce scaling by a negative number as reversing the direction. If a vector is a magnitude and direction, dU must represent a change in magnitude and a change in direction. -1 is a really bad representation of i^2.
Are you referring to the scalar product? In that case, one can consider vector spaces over complex numbers, e.g. ℂ⁻ⁿ and complex-valued inner product. But complex numbers do not apply to geometric vectors as I'm not aware of a way to multiply a geometric vector by a complex number.
@@MathTheBeautiful Geometric Algebra defines vector vector multiplication than results in a geometric object that in 2D maps to the complex numbers. It combines the dot and wedge product, where the dot product is even and wedge product is odd. The wedge product includes an oriented unit multi vector and corresponds with the imaginary component of a complex number. The dot product corresponds with the real component of a complex number.
I absolutely hate the linear algebra approach for defining the vector. Everything is a vector that is a member of a vector space. Thus a scalar and a matrix is a vector, but also a function, or a polynomial. I think it brings the mathematical generalisation to the wrong direction. According to geometric algebra the vector is something with a magnitude and a direction, and this is how people are thinking about vectors. In fact the GA generalisation is more beautiful as well, defining bivectors as objects which have area (magnitude), and orientation, trivectors with volumes, and so on. Yes, I'm trying to evangelise about geometric algebra. 😊
Unrelated to the content of this video, you need to turn off the auto focus of your video camera. Neither the camera nor the whiteboard are moving, so set the focus once at the beginning (verify it's in focus), and lock it so the camera will not change the focus of the lens.
Quick request: maybe you could do a side video reconciling the notion that geometric vectors all proceed from one arbitrarily chosen point (on the one hand) with the tip-to-tail rule (on the other), which requires that they be all over the place? It looks to me that this would require a not entirely trivial construction involving equivalence classes of lines or rays, plus whatever else it takes to define 'sense' in the directional sense. One point of interest is that this would illustrate what it takes to advance from intuition to consistency --- a central issue in almost all kinds of thought.
Good comment! To be honest, I don't see much of an issue. Tip-to-tail is just a set of instructions for constructing the sum and it doesn't apply that vectors can float around the space. But even if you do allow vectors to float around the space - yes, you can introduce an equivalence class where two vectors are equivalent if they are related by a parallel shift. Then you're back to having a simple vector space. The only reason I avoid considering that is that I err on the side of less.
Professor, did you study in France or in a French speaking country? Or have some French colleagues? Because "sens" and "direction" is a distinction I never hear in English courses! Only in France!
Superb channel. Long time fan here
No, I studied in Russia where we only had one word "direction". I discovered the word "sense" in a book I recently read, but I don't remember which. If I rediscover it, I'm come back here and share it.
I certainly was familiar with sense vs direction in my US education. But I'm ancient 😂.
Does the zero vector have “nonsense”? 🤔
@@AdrianBoyko Haha - well done!
You are mentioning homework and exercises in the intro of the lesson; is there a link to them?
When it's ready (enough), it'll appear on my website grinfeld.org
@@MathTheBeautifulThank you!
You can’t introduce dU without complex numbers. Similarly you can’t claim scalar vector multiplication scales the magnitude but then glibly introduce scaling by a negative number as reversing the direction.
If a vector is a magnitude and direction, dU must represent a change in magnitude and a change in direction.
-1 is a really bad representation of i^2.
Of course I forgot the \theta and the 2n in my definition of -1.
Thank you for your interesting opinion!
I hope I’m not misspeaking. But aren’t complex numbers needed to describe vector vector multiplication?
Are you referring to the scalar product? In that case, one can consider vector spaces over complex numbers, e.g. ℂ⁻ⁿ and complex-valued inner product. But complex numbers do not apply to geometric vectors as I'm not aware of a way to multiply a geometric vector by a complex number.
@@MathTheBeautiful Geometric Algebra defines vector vector multiplication than results in a geometric object that in 2D maps to the complex numbers. It combines the dot and wedge product, where the dot product is even and wedge product is odd. The wedge product includes an oriented unit multi vector and corresponds with the imaginary component of a complex number. The dot product corresponds with the real component of a complex number.
I absolutely hate the linear algebra approach for defining the vector. Everything is a vector that is a member of a vector space. Thus a scalar and a matrix is a vector, but also a function, or a polynomial. I think it brings the mathematical generalisation to the wrong direction. According to geometric algebra the vector is something with a magnitude and a direction, and this is how people are thinking about vectors. In fact the GA generalisation is more beautiful as well, defining bivectors as objects which have area (magnitude), and orientation, trivectors with volumes, and so on. Yes, I'm trying to evangelise about geometric algebra. 😊
Thank you for sharing! I love it when people emote over math concepts!
Unrelated to the content of this video, you need to turn off the auto focus of your video camera. Neither the camera nor the whiteboard are moving, so set the focus once at the beginning (verify it's in focus), and lock it so the camera will not change the focus of the lens.
I appreciate the advice! Do you know if there's a way to put the camera in focus before turning AF off?
The full math is revealed when you stop restricting yourself to real scalars.
Thank you for your comment!