Linear Algebra 20g: The Dot Product - One of the Most Brilliant Ideas in All of Linear Algebra
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- Опубликовано: 17 фев 2015
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The Dot Product. Don't leave home without it!
Man, you do a great job. Keep this style of lecturing. Some guys out there need all the small details you are talking about.
I've never seen anyone introduce the dot product like this. I was beaming through this whole video. So exuberant and exciting, thank you! 🤩🙏
Thank you, it means a lot and a Happy New Year!
@@MathTheBeautiful HNY! 🙏☺️🌟🎇
You're one of the very best, thank you for not alienating your audience not even for a second. I've been trying to understand the intuition behind the dot product for yeeeaaars, I think I'm really close now.
I love how you keep saying "Inner pro... eh, I mean dot product". It makes me more intrigued to know what the inner product is (I've never seen it in action). Excellent video!
Thanks! ruclips.net/video/Ww_aQqWZhz8/видео.html
Thanks for agreeing that the Def of the dot product seems arbitrary the first time you see it (even though it might have enormous applications)
Ok. I'll buy 2
The dot (or the scalar) product is the constructed Cartesian bridge between algebra and geometry.
You are a frigging genius!!! I have watched hours of videos and read several articles that would help make intuitive sense of the dot product. You managed to do this in 15 minutes and now I understand it completely! Thank you!! Do you have Patreon??
Hi Timo, thank you that means a lot. I'm glad you found this video helpful. And as a matter of fact, I do have Patreon: www.patreon.com/PavelGrinfeld
Great video. Especially that you tell us why it’s useful
My favorite interpretation is of the product P=AB is P has the same relationship to A as B has to the unit. So the dot product has the same relationship to len(w) as the projection of v (onto w as drawn) has to the unit.
This is so insightful!
@4:30 Thank you! I have been stumped on the "projection of one vector to another" - that made sense with the cos theta.... but why multiply the length? Finally we get an answer
Glad you found it helpful!
You should be the guy to explain functional analysis to me :D
Yes please
Excellent!
I'd like to see this video renamed: "The Inner pr... er,, ah... Dot Product."
The dot product smells a lot like Heaviside. You will find it in the equations he created that we now call Maxwells equations. Vector equations were to a large degree created by Heaviside
This selling of the dot product sounds like an infomerical. "The dot product... it's all you need. It has a million different uses!"
And it can be yours for three easy payments!
@@MathTheBeautiful $i +$j+$k
@@gentlemandude1 That comment is making me see dollar signs.
excellent electromagnetism review
Thanks man, hope you are doing well :)
Thank you, I appreciate it!
dot product or how much two vectors point in the same direction ;).
Thank you so much i was searching everywhere for an explanation like this..id calculate the scaler and think now what? is it an x? is it a y? Wheres the the bit that shows prijection onto 2nd vector..Thanks you cleared it up for me..
The Definition of the Dot Product of two vectors in Rn is UdotV=U1V1+ U2V2+...+UnVn.Then we define the norm or Length(U)of vector U in Rn using the definition of Dot Product.Length(U)=IIUII=Sq.Rootof(VdotV).Then we define the distance between vectors U and V in Rn as d(U,V)=IIU-VII. And only after that, in R2 and R3 we conveniently use law of Cosines for the triangle made by vectors U,V, U-V and derive the formula that you have on the board. That formula on the board is not a formal definition of the Dot Product in Linear Algebra. And the proofs of all the properties of Dot Product in Rn are also based on the formal definition and not the formula on the board. I think it is important from the beginning to buid a solid foundation by using formal definitions and not something that derived later. I like how you connect this important topic to future bigger math concepts and look forward. I really admire and appreciate your passion! Thank You!
Thank you for your great comment! I would only change the very first word: The -> A
@@MathTheBeautiful Thank you. I appreciate that. I never had a chance to learn English. I lived in 4 different countries. So I always ask my students to correct me. And I laugh at myself all the time … Thank You! Your videos are vey inspiring.
nothing I cannot find on a book,
It's more helpful me goods times prefect
Glad to hear that! Would you say the timing was on the dot?
Terrible habit of mathematicians is to describe certain motivations behind mathematical objects and structures as "Something we would like," such as when saying that "Because the dot product is so useful in geometry we'd like to have something like it in other vector spaces."
Well saying it that way makes mathematics seem made up and preferential, and ofc it absolutely isn't
You hit the nail on the head: mathematics is completely made up and preferential. I know you meant the opposite, but the opposite of what you meant is true. Mathematics is the most personal of all the arts, governed by taste and little else. One of these days you'll look back on your decision to leave this comment as one of the most consequential of your mathematical life.
@@MathTheBeautiful I understand that one has the freedom in certain respects to define things as they please, and if the system is consistent then it is at least acceptable. For example maybe I throw out Euclid's 5th postulate and replace it with a new description. I may get crap or I may get hyperbolic geometry. But, describing this process as preferential and made up?
I think that is playing fast and loose with words seeing as we're ultimately constrained by logical consistency. But, Im willing to be convinced otherwise as you say maybe this is a turning point in my view of mathematics.
My point though was that just because something is useful in geometry doesn't mean you'll be able to find an analogue in linear algebra. So is it just that you want there to be an analogue so you begin a search for it?
As much as I try to avoid this debate, I can't. 😂 It's personal and most of us have an opinion/belief on it. How about.. it is completely made up, and preferential, according to rules we decide to agree on? Collective preferential consistent nonsense that is useful?
Maybe what belief we hold here is what we personally find empowering/inspiring. For some of us, it is an objective reality that we can possibly know. For others, it is freedom from such a reality, whether or not that freedom is real.
To much blah blah. Please go to the point
The guys must be from senior management: mission, vision, value proposition, value statements... bla bla bla for 15 minutes and finally arriving at u = v multiplied and divided by w^2... which kinda tell that u = v (...perfectly interesting geometric quantity... bla-bla-bla again... WTF?)
When you reread your comment in a few years, you will disagree with yourself.
MathTheBeautiful dont get offended please, I found your other videos on the subject and those are much better structured, thanks!
"InNeR PrOdUcT" in other words, let me keep using this term purposely to make myself seem very smart and above everyone to feed my weird ego. No one messes up that term that much its so obvious and cringe.
This guy doesn't own this topic