What is a Vector Space? (Abstract Algebra)
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- เผยแพร่เมื่อ 19 ต.ค. 2016
- Vector spaces are one of the fundamental objects you study in abstract algebra. They are a significant generalization of the 2- and 3-dimensional vectors you study in science. In this lesson we talk about the definition of a vector space and give a few surprising examples.
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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
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Milne, Algebra Course Notes (available free online)
www.jmilne.org/math/CourseNote...
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You explained in 7 minutes what my professor couldn't explain in a whole semester during two hours lectures.
But she might be learning it in a way like you did: hearing those boring lectures. Yes, sometimes it is, but by that a lecturer wants you to start doing something by yourself - to put more efforts.
@@philosophyversuslogic Nah. It's just hard to get people who are really good at somethng AND also good at explaining/teaching it, WITH the inuition to understand the pain points from the point of view of the student. Truth is students are given textbooks, boring lectures, teachers who don't have the time or ability to explain things, and no time to properly learn. The vast majority of professors are simply doing a job they must do (ie. get through x amount of material in y amount of time). You are excusing their utter dogshit "teaching" practices. Yes, you absolutely need to learn on your own, but having a bad teacher with poor explanations WAISTING YOUR TIME is FORCING you to learn on your own in a way that is not just unoptmial, but harmful to the student.
Imagine theres 2 scenarios: #1 Student sits in a 2 hour lecture explaining 10 concepts in which 10% is retained (1 concept), ends up spending the same amount of time if not more trying to understand the remaining 9. The student is confused, and thus must spend an exhuberent amount of time trying to understand. #2 Student sits in a 1 hour lecture in which 2 core concepts are explained WELL, and 8 oncepts are very lightly brushed up on, with recommendations/material offered for the student to study outside of class.
Thank you so much for making it easy and simple to understand, and yet not "talking down" to any new students of the subject.
Honestly this channel sets the standard for online education resources.
Easy and simple. Ha. Way to repeat yourself.
I find you funny and hilarious.
So easy, y u no da study and no eh prus u failure. Ai yah!
One of the few who don't give a text book definition but actually try to explain the logic behind it.
Simple, poignant, and well-executed. This is an underrated channel that can easily make advanced mathematics accessible.
You are so kind, thank you.
This is truly our goal - we love math, and we think everyone deserves the chance to learn.
We would really appreciate it if you would share our videos with your friends - on Twitter, Reddit, etc.
That will really help us grow!
Thanks again for watching and for leaving such an encouraging comment. It really makes us excited to make more videos! :)
@@Socratica appreciable
@@Socratica how about some content on homology groups? Eg. Mayer Viertoris sequence etc. I find that stuff difficult to understand.
I swear to god, I am in love with this channel. Linear Algebra is NOW MADE EASY.
I like that socratica chooses to cover abstract subjects.
indeed!!
@eh6794 Vector spaces and modules in pre-calc?
@eh6794 Pre calculus lol, in what university?
@Artificial Resonance How old were you? Was it a private school?
@Artificial Resonance No you didn't lol, this is not part of any European high school curriculum. You probably learned about vectors and operations on vectors, and can't tell those apart from the algebraic structure that is a vector space.
Arguably the best and simplest explanation of a vector space I've ever heard. Thanks a bunch!
Thank yooooo!!!! I'm taking linear algebra with a midterm test in 2 days. The book throws all these rules about vector spaces but never does it take the time to explain or help visualize WHAT a vector space actually represents. I can memorize the rules but I just felt so lost.... But this really helped me.
We're so glad to hear this helped! Visualizing math/physics is one of the best uses of video, we think - it really adds something to your textbook reading that's hard to do just in your head!
Good luck on your midterm!! :)
I love this channel. I went back and looked at some of the earliest videos and the progress is amazing. You've come a long way from a 1 minute video on a powerpoint slide. This is one of my favorite channels when I need something complicated explained.
That is so kind of you to say, thank you!! It has truly been a labour of love. We are completely self-taught filmmakers and editors, and it's been a real education! :)
Took a "complex" thing and made it simple and clear, thank you
A physicist and a mathematician are in a congress and the talk goes like:
"If we consider an 8 dimensional sphere, we can show that the radio..."
The physicist, skeptical, ask the mathematician:
- "Do you believe such nonsense about 8 dimensional spaces?"
- "Sure, of course"
- "So.. how do you "see" an 8 dimensional sphere?"
- "Easy. I imagine a N dimensional sphere, and then set N equals 8".
While a theoretical physicist is simply enjoying the debate
An engineer looks at them as looney
@@goyonman9655 Theoretical physicist leaves the chat for PEACE!
In my experience, physicists know the difference between mathematical space and physical space.
@@lauterunvollkommenheit4344 wrong.... "physicists" is a very general term... when u say "physicsit" do u mean experimental, theoretical, mathematical...depending on which physicsit u are, u'll understand and interpret math stuff differently. Cheers!
I have an exam in three days, and you just helped me understand vector space better than my teacher and textbook. I feel more comfortable with the topic and have a better grasp of it. Thank you!
You are the maths teacher I have been looking for. Thank you so much socratica for helping many of us to imagine the concepts from maths.
This is an incredibly helpful video, I was unsure of what a vector space actually was but this video explained it ridiculously well. Thank you.
I can't thank you enough for this. Your amazing videos make my university lectures so, so much easier to digest.
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The "ow" and the pointing vectors were so cute. Makes me wanna hug this lady.
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This is awesome stuff, Socratica!!! It's very, very easy to become a huge fan of abstract algebra watching videos like these!!! :) :)
Superb concise lucid presentation! Good writing, splendid narration, + graphics and simple practical examples make this easy to follow.
Style of explanation is awesome. I like the way you touched the arrow at the end.
It has become my favourite channel. Concepts are nicely explained.
I love your channel. Your presentation is clear and creative and polished. I often learn new things here, and sometimes I understand things I was "taught" in the past, for the first time.
I'm speechless. Such a clear and precise explanation, really makes me happy that I stumbled upon your channel! Big thanks from a student who was facing a hard time understanding these abstract concepts. I'm going to watch all the videos related to my syllabus, that has been covered by you guys. Peace!
This channel is a hidden gem. Thank you so much.
I'm New here and as an Electrical & Electronic engineer student i like your way of explaining thing , work hard for more great content ( you already earned my subscription)
Vectors addition, multiplication rules can be a applied to n-dimensions. In Computer Games like Age of Empires 5 dimensions are Food, Wood, Gold, Stone, Population.
AoE is the bestest of the best. Thanks for making this interesting connection.
You mean Age of Empires ll, right!
More abstractly I tell my students that a vector is just an ordered set of elements
{a,b,c}={b,a,c} is a set but (a,b,c) =/= (b,a,c) is a vector.
Thank you so much for making the concept of vector space and vector definition extension explicit with your perfect teaching.
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This might just be the best TH-cam channel on mathematics! Great lectures!
after al this definitions it would be nice to see their applications to proofs
woow. I am so lucky to have found this channel. Its like a gold mine here.. such awesome explanation! And highly immersive for a dry topic when studied from text books!
We're so glad you've found us!! 💜🦉
This channel has got a new subscriber today.
Just amazing explanation. Keep making such useful videos
Awesome video! also left a smile on my face at the end...U r really helping students explaining stuff books don't (even my prof didn't explain it this way)
The sound effects make it sound so sci-fi. Awesome.
This is an excellent demonstration of teaching at a comfortable rate with succinct information that is easily digestible.
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Excellent explanation in a short video. Bravo
You just saved me hours of research!
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Dankeschön für die Auffrischung. Sehr verständlich erklärt.
very well explained, great video! keep going on
This is spectacular.... You've actually explained it very well and I like it
More abstract algebra please!
I understood a 2-hour course in less than ten minutes. Thanks
fluent to the point and distinctively i like it
In addition to a beautiful presentation, the presenters carefully select their attire and background.
To me, this is extremely calming to the eyes and facilitates the brain's storage of the concept.
This is the best explanaition ever. This video most be the first one. No other one explains much better than this. New sub
Rarely do I subscribe to a channel just by watching one video, but this time I did. Thanks for the high quality video!
I would have to say that finally I understood what is a vector space. I wish someone would have taught me like this. Needless to say I scored a B+ on Linear Algebra and don't know what is a vector space ! You are great ! Keep up the good work !
A B+ in math? Ooh I hope you don't have Asian parents
i've taken vector spaces 2 years ago and finlay i understand it this day. thank you so much
I understood them right from the beginning in lecture. They are not that difficult.
your lectures are awesome! So much better than my lectures at school
Honestly, this helps a lot. Thank you very much
Very engaging teaching and presentation style. Have finally grasped the concept. Thanks :)
Thank you sosoooo much. This is an underrated channel.
Theoretical Physics only goes so far, but theoretical mathematics has no boundaries. As physicists climb the prongs of theory, we will start needing more tools to utilize for the purpose of measurement and evaluation. You're awesome! Thank you.
Very nice and easy to understand explanations!
honestly love your channel! it helped me understand this thank you !! just subscribed
We're so glad to hear this! Thanks so much for subscribing - we're so happy you're watching! :)
That was a really good explanation
This series on abstract algebra is the best thing I’ve ever seen
You're so kind, thank you for this encouraging message! 💜🦉
@@Socratica Thank you! I wish you had a Linear Algebra series too!
It's just really damn awesome. I learned vector spaces in a way that i could never have imagined. Really impressive 😍. Thank you ❤️
Explained very well....Thank you.
By far the bestestestest of all the abstract algebra lectures in this world..
This teacher is ideal. The lessons are not only informative, and highly intellectual, they are lovely!
Good job.
Really nice quality videos.
I JUST BEGAN TO FALL IN LOVE AGAIN WITH ALGEBRA. IT LIKE SHOWING ME SOME SCIENCE OF QUANTITIES IN THE UNDERGROUND. SO SWEET.
Is zero vector a starting point? It’s been a wonderful teaching that you give thank you for creating this
I watched many videos on vector space but i did not understand... But ma'am you explained very well and in better way. Thanks
Thanks for this video! I was interested to see how vector spaces could be things like the set of continuous functions and polynomials of order 5; I thought of them very much as space-time constructs (speed, distance, acceleration etc.).
Before taking first upper div (linear algebra) course, this is video is strongly recommended. I believe most instructors can't teach as clear as this lady.
Most awesome teacher of higher mathematics...thank you for making maths easier for us...
really a great and very clear explanation, thanks
Very nice presentation ! Bravo !!!
Keep it up I love this. Do Galois Theory next :DD
which physics class are you taking
Lol
Well done i am always confused in the vector space you explain it very well
Absolutely fantastic...finally I got it
1hour College class = 6 minute video fantastic Job thaNks Socratica for that effort
Thank you so much you help me to understand linear Algebra you are brilliant God bless you
First video I watch, you got yourself a new subscriber!
Outstanding presentation!
Thank you so much . I found this video VERY helpful.
This is the best presentation there is.
this was awesome and the ending was just perfect lol
Your explanation is great...I like your channel..it helps a lot in studying algebra..pls make more videos on modern algebra and other math topics also.
Phenomenal teacher! ♥️
Amazing.Do more videoes.You are very good at explaining the things.
Socratica has some of the clearest explanation out there
beautiful channel and explanation of concepts... regretting to discover it so late..still very helpful
We're so glad you finally found us!! 💜🦉
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Osm, now I cleared my doubt regarding vector space in linear algebra.
Thank you ma'am.🙏🏻
Great explanation! Thank you very much !
The idea of pointy vectors is awesome.
hi socratica ur explanation way is really very good . Hope that some day our teachers will also learn this kind of way of teaching
The way to explain things very easy and simple,which I like it. thanks
This videos are the best, the las year i saw a course of Groups and Rings, and was nice watch this videos and remenber those classes, thanks for the amaizing work in this videos i am waiting for more, a big hug from Colombia!
Great video as always!
Just amazing...... got idea of vector space ......... very broadly the topic is explained ,at the beginning I was thinking vector space are just vectors .......thank you.......
So amazing explanation... thanks
Great effort! I really enjoyed your videos as I am trying to teach my son math in the middle school.
i seldom press the like button for videos on youtube but I make sure that I do it for every video I watch on Socratica ;)
I need to comment just because the ending, that was pretty cool :)