What is a Vector Space? (Abstract Algebra)
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- Опубліковано 28 вер 2024
- 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/...
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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!
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.
@@alanveiga452 yes bro
fluent to the point and distinctively i like it
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The "ow" and the pointing vectors were so cute. Makes me wanna hug this lady.
How to get blue shirt like you
Took a "complex" thing and made it simple and clear, thank you
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! :)
This teacher is ideal. The lessons are not only informative, and highly intellectual, they are lovely!
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 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 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'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!
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.
Is zero vector a starting point? It’s been a wonderful teaching that you give thank you for creating this
This channel has got a new subscriber today.
Just amazing explanation. Keep making such useful videos
Thank you sosoooo much. This is an underrated channel.
I love your teaching 😍 it is not easy to focus on studies when you have financial crisis .I can focus on the study without distraction in your UA-cam channel. Thanks
Superb concise lucid presentation! Good writing, splendid narration, + graphics and simple practical examples make this easy to follow.
One of the most amazing example of vector spaces is the one that embeds words (that is, a function that turns a word to a n-D vector). When you embed words as multi-dimentional vectors in such a way that vectors that are close together (that is, vectors that have similar norm and close to "angle" are close to 0º) means that they have semantic proximity you get to do vector arithmetics with words.
That is, if you start if vector for "king", subtract "man" and add "woman", the result is vector that will be pretty close to vectors corresponding to "queen, princess, etc.."
These vector spaces dimensions often hit the mark of hundreds.
It was so clear and interactive...I am a Masters students and I was trying to understand this topic from a so so boring lecture of a professor.....Thanks to this team atlast I could get on it :)
What do you study? You're a masters student but can't understand an undergraduate level concept?
Socratica has some of the clearest explanation out there
hi socratica ur explanation way is really very good . Hope that some day our teachers will also learn this kind of way of teaching
1hour College class = 6 minute video fantastic Job thaNks Socratica for that effort
Nice madam prepare more lectures so that people get more benefits and knowledge from ur skill,experience and knowledge.ur way of teaching is outstanding
One of the best teacher.
It's just really damn awesome. I learned vector spaces in a way that i could never have imagined. Really impressive 😍. Thank you ❤️
Keep it up I love this. Do Galois Theory next :DD
which physics class are you taking
Lol
First video I watch, you got yourself a new subscriber!
Great effort! I really enjoyed your videos as I am trying to teach my son math in the middle school.
Just a small technicality, but forces aren't vectors, but co-vectors. This is because it is a linear map from a vector space to a scalar. Illustrated most clearly by 'power', where it maps velocity (vector) to power (scalar). Hence why it is also represented by a row and not a column vector. It is the dual space of the vectors (velocities).
I'm working on 3d algorithms for my game. After the past 2 hours of getting more and more confused down the wikipedia rabbit hole, this video cleared everything up in 7 mins 🙈 😂. Thanks!
We love hearing how people use our videos in their work! Thanks for telling us!! 💜🦉
I need to comment just because the ending, that was pretty cool :)
Most awesome teacher of higher mathematics...thank you for making maths easier for us...
The crux concept of space in mathematics revolves around the concept of closure. Whatever mathematical entity we define, its algebra must produce the other entities that exist in the same space or less. That can’t land outside that space. So, if we take two vectors in 2D, the algebra we use on those two vectors is limited to the rules that guarantee we don’t land in 3D or higher dimension. So the 2D space is the collection of all vectors along with the operations we could perform on them as we define such operations as long as the output is also a 2D vector that lands us in the 2D space. For instance, if we define an operation on vectors that produces an output in the form of 2by2 matrix, we would not stay in the 2D space. So that operation violates the vector space definition. Similarly, real numbers are a vector space with some operations defined such as addition, multiplication but not with a square root operation, because taking the square root of a negative real lands us into the complex place. And so forth.
your lectures are awesome! So much better than my lectures at school
Never stop making this videos! Oh my GAWD i love them so much!
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! :)
AMAZING JOB
Madam your teaching skill is very nice please makes more videos on the calculas and algebra
Very nice presentation ! Bravo !!!
too simple to understand
the best video I found for vector space..!!!
yess
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study math they said, no harm can come to you they said - lecturer immediately gets attacked by vectors. I'm done back to shop class where it's safe.
I request that make a video on norm, difference between norm (||.||) and absolute(|.|) and different types of norms, such like (||X||1,||x||2,||x||inf,||x||P) norm. Thanks
This video was soo much more useful than Khan's! Well done!
yes
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Osm, now I cleared my doubt regarding vector space in linear algebra.
Thank you ma'am.🙏🏻
My prof sounds like he smokes crack before every single lecture. It is a nightmare. Thanks for a comprehensive video.
The Mathematics video series provide by you are really top class.
Would you please present us video series on the following topics :
1. Tensor Analysis
2. Non Euclidean Geometry
3. Topology
Thanks
"They" are vectoring in on my position right now!
Just finished my second time watching these abstractly amazing videos. When you guys are releasing more videos in this series?!
A = {a_i | a_i = ith video in the Socratica Abstract Algebra playlist}
Theorem: ∀a_i∈A , a_i is awesome!!!
We're so glad you are enjoying our videos!! We're busy editing an Abstract Algebra video right now! It should be out tonight. :)
Socratica So excited!! Thanks for the reply.
You’re gona win a Fields Medal for proposing this theorem, lol.
Thank You Mam
Nice video plz cover linear algebra
If a scalar can be a complex number, I am trying to imagine what the resulting vector space looks like. Its members will also be complex.
Of course. Scalar can come from any field.
Thank You Soooooo Much Miss🙏❤️❤️
excellent teacher
amazing lectures indeed. really a great effort thanx socratica
You are great. I really like your presentation.
It's so amazing. Thank you so much :)
Hi ! I'm Victor the Vector!
Very nice video, but I would mention (especially in the light of the last few videos) that the definition of a vector space does not necessitate action by R but by any division ring...
The associativity property is typed incorrect. c x d should be c . d, not vector product but the dot product as in the left-hand side of the equation.
Someone might have already pointed out that already, but alas...
nice explanation!!!
very interesting method
thank u madam...u make maths a fun...
Right 😁
can you please do a video on Hilbert space ?
Dear madam ,u r knowledge is beyond the praises thank u
But will please give me abstract defination of multiplication?
Right from my childhood I have this doubt
When we multiply exactly what we r doing with it
Plzz reply as soon as possible
Thank you
Your Channel is just fantastic!!!
last prt was really good
i love Socratica
Channel!
So good explanation
In Age Of Empires, if Food = 50 Wood = 20 Gold=90 Stone= 10 Population=7.
5-Dimentional Vector is A=[50, 20, 90, 10, 7] & Length of Vector A = Sqrt (50x50, 20x20, 90x90, 10x10, 7x7) = 105.589
Length of n-dimension vector square root of dot product of vector A with vector A
Plzzzzzzzzz plzzzzzzzzz make videos on Riemann Integral and also probability.....plzzzzz❤️❤️❤️❤️❤️🙏🙏🙏🙏🙏 I love your videos so much your videos are so much helpful
What about the existence of a zero vector, additive inverse, and closure under vector addition in the definition?
Thanks. Could we multiply a vector with a complex number too. Also could you tell me how Dirac used it to modify the Schrodinger's equation to include the spl theory of relativity?
r
Thanks a lot madam. it was very useful for me
Thanks!
Very informative... Thank you!
Hi,
Thank you for the video. The operators such as dot and cross used between the scalars have no meaning as it would have been, if used for vectors? Am I right?
When you say inverse of a matrix, you mean additive inverse. Am I right? Cause a matrix at least has to be a square matrix even before we think of it's inverse(of course multiplicative).
This host is the exact female version of the host of the UA-cam channel PBS Space Time. Uncanny!
It's the hair and jaw lol
nice explanation
thanks for your amazing video , but i have equation . you said 2*3 matrices form of commutative
group which is contrast with ( matrices are non abelian group)
You have to pay attention to the operation. 2x3 matrices form an Abelian group _under addition._ Matrix addition is commutative, even though matrix multiplication is not.
thank you madam.
Nice explanation..
Thank you...very good.
Error: At 3:26 there is a cross product instead of a dot product for the associative principle.
It's not an error. They used different multiplication symbols to distinguish between multiplication of field elements and scalar multiplication of a field element on a vector. These are technically different operations. We often use the same symbol to denote both, for simplicity and laziness, but it is nice to keep in mind that these are technically two different operations.
is vector space is not only the collection of vectors satisfying the certain axioms??how can a collection of matrices,polynomials,functions etc ,be a vector space??plz explain it.
this much info in 7 minutes.. astonished O_o
Can you giv us a video on Sylow subgroups and its theorem
Loved this viedo
What is difference between vectors and vector space? kindly Ma'am
Impressive =D Thank you!
Make more vedios on vector space
If i had a teacher liken this i'd take up physics as a major
thanks alot very helpful
Wonderful!
Superb
For matrices I wouldn't confuse inverses with negation if I were you
Thank you ma´M
This is awesome
if adding a constant to a vector instead of multiplying a constant, is the constant treated as a vector instead of a scalar? (in the graphic version of this, translating up or down instead of scaling longer or shorter)
While you can multiply a vector by a scalar, you cannot add a vector and a constant. For example, 2*(1, -3, 4) = (2, -6, 8). For addition, however, (1, -3, 4) + 2 is not defined. You can translate, however, by adding a second vector: (1, -3, 4) + (0, 2, 0) = (1, -1, 4), or even (1, -3, 4) + (2, 0, 0) = (3, -3, 4).
@@Socratica Isn't there an imposed addition function meaning: If we have a vector such as {3,5,7} and we have an + operator on that vector where the other operand that is scalar N, the full operation can be represented as {3+N, 5+N, 7+N}? This would be defined more in the field of Affine Transformations within Linear Algebra as that Scalar would scale each coordinate component by that scalar quantity. This would apply scaling in all directions of that vector space. I've written C++ mathematical libraries consisting of vec2, and vec3 objects and a matrix object that is a 4x4 matrix. And using operator overloading of those vectors, I was able to apply the scaling or shearing of those vectors within the 4x4 matrix. I think this might be outside of the concept of Abstract Algebra but would fall under Linear Algebra and Vector Calculus with Analytical Geometry. Just curious to what you think... I appreciate all kinds of feedback and positive criticism is always welcomed!
How about integer values matrixes? Or vectors over Zn? They form commutative group, but your slide does no put any restriction on the scalar field F. Can a scalar be complex then?
As long as the scalars come from a fixed field and the axioms are satisfied, then you're good!
So, for example, your field could be the complex numbers. Then you would have a complex vector space (or C-vector space). But you have to make sure the axioms are satisfied! If you take nxn complex-valued matrices, this is fine. But if you try to restrict to nxn real-valued matrices, then this isn't closed under complex scalar multiplication. (For example, i times the identity matrix is not real-valued.)
By the same token, you cannot restrict to integer-valued matrices and have it be a vector space. Non-integer scalar would scale matrices out of the set.
But this naturally leads to a really nice question. If we wanted to consider integer-valued matrices, what set of scalars would we need in order for it to be closed under scalar multiplication? Well, the integers would work! But the integers aren't a field! So this would not be a vector space. But, the integers form a ring! And that's good enough for people to study it. Structures that are like vector spaces but where the scalars form a ring are called "modules". Socratica even has a video on modules!
Good one! so, simply put, rule 3 puts restriction on F. Thank you for the extended explanation!