Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra
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- čas přidán 5. 08. 2016
- The fundamental concepts of span, linear combinations, linear dependence, and bases.
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I don't know who you are, but these linear algebra videos are brilliant. They are pedagogically invaluable and should be incorporated into every introductory linear algebra course. I teach linear algebra and I mention these visualizations but my hand-drawn figures on the marker board, my clumsy gestures in the air, and the textbook's static graphics are all quite inadequate for most students. I will be directing my students to these videos in the future or even playing them in class. Thank you.
I know you wrote this a year ago but as a tip, my Maths teacher often took pens to show vectors in 3D space and sheets of paper to indicate how the plains would cut. I think this visualization is important and also a good method to engage students to think about how the graphs would look without always drawing them first. The transition between 2D to 3D was hard on some students and they just couldn't visualize it when drawing on sheets of paper. This helped a lot and was fun, I still do it sometimes.
Thanks for your contributions to educating the public :)
*pedagogically*
He is a professor. That Phd don't come cheap
Very nice!!! Thanks a lot
I'm an astrophysics phd and I use linear algebra everyday but I'm here watching these videos because they are so intuitive...
true....i'm too phy phd from harvard but i like these videos too...its a good source material to give for the students.
What do you study and/or research?? I am a high school student very interested in physics and mathematics and would like to know
@@shreyansj2703 these types of videos are useful to harvard students ??
@@shreyansj2703 harvard is that community college right?
@@ergpopler413 nah I think it’s a middle school
I am still in 12th grade and this is all I had on Google Play rewards money. But as soon as I come of age(later this year), I will save up money to send you dude. Stay at it bro
Seriously, the way he explained visually blew my mind. I was having such a hard time visualizing all the terminologies; I could understand them based on the definition in the textbook but was getting such a hard time having a clear mental image. I can't thank him enough. I would also donate, once I get a job.
based
1. a coordinate is a scalar, which scales the basis vector of its coordinate system
2. anytime we describe a vector numerically, it depends on the basis vectors we are using
3. a linear combination of a set of vectors is to scale them and add them together
4. the span of a set of vectors is the set of all linear combinations of the vectors
5. if a vector is a linear combination of a set of vectors, the vectors are linearly dependent
6. if each vector adds another dimension to the span, the vectors are linearly independent
7. the span of two linearly independent vectors is the 2D space, if the two vectors line up, their span is a certain line
8. when thinking about one vector, think of it as an arrow, when thinking about a collection of vectors, think of them as points
9. in three-dimensional space, the span of two linearly independent vectors is an infinite flat sheet, the span of three linearly independent vectors is the 3D space, if linearly dependent, the span is still a flat sheet
10. the basis of a vector space is a set of linearly independent vectors that span the full space
thanks anime man
Thanks!!
❤
You can generalize point 9: the span of a set of vectors is a subspace of a number of dimensions equal to the number of linearly independent vectors in the set
7. ...if the two vectors line up, their span is a certain line, and they are linearly dependent
Found this series 5 days after taking my Linear Algebra final. It's nice to finally understand what was going on the whole semester lol
Moood
duuude lol!
I'm studying for my final now and am embarrassed that up until now i have had no idea what a basis and span actually were. They are such simple concepts but were explained to me in an unintuitive way. So glad i found this channel!
totally a savior lol
You bro are a legend 😂😂
So many years of 'rigorous' linear algebra, but I still didn't have a good understand of the intuition behind it. Grant, you are a miracle worker. So happy to see you ended up in the math education field! fsc
FSC!
Same here. I'm really impressed!
This is (understated) the most intuitive that algebra can get. Its like I am learning again and can teach my niece when shes in high school :-)
What is the difference between rigorous and vigorous? Doesn't the latter seem more friendly?
rigorious 😂 😉
Thank you! This is probably one of the most beautifully explained videos ever, your voice and animations are incredibly helpful to understand and enjoy the video👏🏼💐
I'm just gonna binge this tonight instead of netflix haha
@Priyanshu Guha unless you’re watching Dark
@@citrus4419 Dark is so underrated maaaaan, its such a coincidence that people in this comment section like it too
@@citrus4419 es wird weider passierein
My turn
Complex Algebra: "We call the vertical unit i."
Linear Algebra: "We call the horizontal unit i."
Every person in the universe using cartesian coordinates: Horizontal is x.
Surveyors: Let's say that's y instead.
@@Renisauce Are you for real serious??!
But it has a hat in top!
@@master1900mc big brain
@@master1900mc Statistics: Let's use a hat to denote the idea that this is a predicted value, not an actual value!
I literally left my Cornell math support tutoring crying, feeling worse, but some girl stopped me to direct me to your videos. Thank God for her and for your videos, bc I was on the verge of a breakdown
aww bless her
fucking cornelll doesnt even teaches properly!?i thought it was only my college
Cornell: "All your basis vectors are belong to us"
@@syedmohammadhussain7137 Just bc I didn't understand doesn't mean everyone else didn't haha
Uh oh I’m taking 2210 linear algebra next semester
"as you scale that new third vector, the sheet moves through the entire 3D space”, that’s the kind of thinking that pushes people to understand the concept for themselves. Thank you so much for your videos!
MIT undergrad here. Your video just taught me in 9 minutes what my math professor and teaching assistants couldn't in the past 2 weeks. You're amazing thank you!!
I don't know who you are, but I will find you and I will thank you
This is him : www.3blue1brown.com/about
It's supposed to be a reference to Taken, Ed Ed.
@@lenkapenka6976
How does that have anything to do with being Indian?
czcams.com/video/Xcz-rVPvL2Y/video.html
That's you when you meet him
@@Masardirasa i like people like you hhhhh
The peaceful music really helps set aside the onset of anxiety that usually comes at sight of numbers and equations.
Thomas Thomas I watch one of these videos before going to sleep. Love the music at start.
you'll love this album then:
vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
but why numbers and equations make me calm?
That part makes the video even better
I’m a geek so the sight of equations is what gives me peace. What’s funny is that I’m here because I had a tough day and I need to relax lol.
4:00 span of two vectors
6:50 two vectors in different plane. your span is the new plane
7:48 third not on the same plane as other two (ie their sum) means it unlocks every 3 dimensional movement in that third v direction
8:33 linear dependence
Lets get this up voted!
thank u
I'm a school dropout and I could never understand math. I thought I was stupid and had no talent for it, until I found this channel, which to my surprise helps me understand! This is better than any math book I've ever tried to conquer. The video format bypasses my mental block which interferes when I sit down with pen and paper. I feel like these videos are teaching me the general process of math and its nature of problem solving, so with this I can finally learn to self-learn.
Education has made some amazing advancements, and (at least my local)school system seems to be lagging behind. I have never learned from teachers and homework, and writing with pen and paper. There must be more people like myself out there that need to be shown that there are alternate methods to learning which might suit them better.
I study at one of germanys top engineering universities and your way of explaining this is so much more superior than my univeristy professors.
Aachen? Xd
KIT?
I am also studying at a "Elite" University in germany and the quality of the lessons is abysmal. These videos are not only a life saver but also sparking a love for math that i never knew i had in me.
Haha same.
TU Munich or KIT?
The definition makes sense since a linearly independent vector (no matter how matter how many dependent vectors there are) unlocks exactly one new dimension. 1 vector can describe all of 1d space with a scalar of some kind, but only 1 space. Adding a linearly independent vector of that one unlocks another dimension and so on. If we were to add a linearly dependent vector we would not get a new dimension no matter how we scale it. awesome video btw (i hope my comment was readable)
I am trying to understand the question as related to what he described about the basis at the beginning of the video. At the beginning, he described when we have two vectors, with a linear combination of two vectors, each scaled with one independent scalar, the entire coordinate system with a dimension of two can be captured by just those two pairs of vector and scalar. The span is the entire space of this coordinate system, and the linear independence is simply saying that those two vectors can do this job of capturing the whole space. Then when we think about what are the conditions to make them linearly independent, we see that they just can't be described by each other. When thinking until this point, I just realize that linear independence is probably the most efficient way to expand the space of possibilities! It's like when two people are linearly independent, they can achieve much more possibilities than if they are not linearly independent, including nonlinearly independent and linearly dependent, with decreasing possibilities! (Trying to find someone in the comments that answers his quiz and force myself to think. Thanks for the comment!)
Couldn't a vector (x,y) reach every point in a plane? Why do you need two vectors?
Jearl Price consider a 2d plane with only one vector, we can scale it however we want but it wont reach all the points in the plane, it will just create a single line across the origin
Kristoffer: I think i see what you mean but you must be careful with the definitions. Remember that a linear combination of a set of n vectors, call the vectors X1, X2, ... Xn, is a vector Y = aX1 + bX2 + ... + cXn, where the coefficients are real numbers; that is, it is the result of scaling all of the vectors by arbitrary real numbers and then adding them together. If we have just one vector, call it X, then a linear combination of X is Y = aX, where the coefficient a is a real number. Now the span is the set of all linear combinations of the chosen vector(s). So if we just have one vector, again say X, the span is the set {Y = aX : a in R}, which is just a straight line (since scaling a vector can at most reverse its direction; it can never break out of that line). We must have another vector that is not a scalar multiple of X to break free from this line and span the plane. I hope this has helped and isn't confusing. It would be easier if we could embed Latex or somethign like that in our comments to make the notation a bit more clear.
It can but here we are talking about how many points in the space we can reach using the vector operations. If you were to just simple add random points and keep adding them in the ordered pairs (x,y) you can get any point in the space but you are not doing any operation here.
The basis of a vector space is a set of linearly independent vectors whose linear combinations can span the whole vector space.
After 12 years,I realised why it's called linear algebra!! Hats off to you sir..
@Thomas White what
what
what
What? He specifically said it wasn't the actual etymology though...
Did you actually look up the actual etymology?
i hat or j hat?
These videos are a gem.
They must be preserved for all eternity!
68th like ...nice
I agree
@@Gravitraxer_AangCZ 169th like …nice
Thats what I was saying right now! Exactly this!!!!
You managed to explain in less than 10 minutes what my professor failed to explain to me for half a semester. All linear algebra students should be required by colleges and universities to watch your videos because your videos just cut to the important stuff without any unnecessary BS.
I love 3b1b videos too, but unnecessary BS is needed for rigorous math. Obviously, not all people want to know rigorous math, so your comment is totally valid.
@@zokalyx I would like to add to this. What he's referring to as "unnecessary BS" is what visually thinking people would usually think of math that is only taught with definitions and logic, with no visual understanding to reinforce it.
There are some that learn better with that method, but others that learn better visually. This is why I think that many math courses should have a visual intuition version and a logical version. Both teaching the exact same thing, but in a different way. Then students can choose which one they'd do better in.
I'd imagine that at least 50% of people who say "I'm bad at math" only say that because they weren't taught it in the way that was best for them.
@@shipwreck9146 To even futher expand on this. I think it's wonderful to start with a good intuition before diving into mor rigorous math. However, the logical version is really necessary to understand more generalised statements, very intuition can certainly guide you but can fail you as well or might be outright impossible.
@@hungdo6397 That's also a very good point. After re-reading my comment, I should have specified that you can't learn everything from math just by seeing it. But rather that by seeing it, it reinforces what you're doing with the math.
@@shipwreck9146 Definitely agree! For example having a firm grasp and intution in Linear Algebra certainly helps understanding Functional Analysis, but relying solely on intution will fail you there.
I would like to leave an appreciation for the fact that you start with something most students are familiar with, develop our intuition and finally provide the definition. School teachers please learn how it's done. I cannot stress enough how helpful your videos are, thank you! Greetings from Portugal
Holy crap. I have been hammering through my textbook and lecture notes on spans, linear dependence/independence, and basis, and I feel like I’ve had my mind blown by the intuition you gave me by showing the math graphically. Everything makes so much more sense. Normally I don’t comment like this on other tutoring videos but this is soooo helpful
This channel is so damn underrated
Agree ! But 1.2 million subscribers for a Math channel, not bad I would say.
Math is underrated :(
@@coffeedude underrated comment.
Hi I'm future, don't worry he got 3M subscribers now it's not underrated anymore
@@random-0 He needs more subs then T series and he still would be underrated, it's gold.
Best material on linear algebra.
Thank you so much.
Never could have known linear algebra without you
I want all my mechanical engineering brethren to watch this whole series. 6 years and still the most intuitive linear algebra material I've ever come across.
The lecturers on the linear algebra course at my university recommend this series as supplement. And I see why. It especially helps me to visualize what's going on.
Today's generation is blessed to have this channel.
im actually screaming this made so much sense like i think i get it now my mind is blown
As a 41 year old software engineer I thank you for this outstanding playlist allowing me to understand things in a new, more practical and visually stunning (as well now visualizing) way!
One 10 minute video explained the basics of Linear Algebra better than an entire semester at college. Thanks Grant!
all linear algebra lecturer should teach us like this so that student can get the picture about it ,not just read the definition.
I agree, I wish they did ... but since the educational system has not changed since the big bang than the chances are very small that it will be applied for the future generations. #sadtruth
AOJ keygen I very much disagree, I think it's wrong to think about vectors in the geometric sense in the beginning, you need to start from the abstract to the geometric examples, otherwise you won't be able to think about the abstract cases correctly.
Better yet simply replace all of these leeches who, regurgitate the same poorly thought out material every year, with khan academy and this dude.
+ועי איליה
I couldnt disagree more. Most people arent smart enough to deal with very abstract topics from the getgo and it all just boils down to a series of uninteresting operations that they forget as soon as they pass the course. I also dont understand your point about learning geometrically first being a crutch when moving to the abstract, that makes very little sense. People like seeing the real world aplication before they move onto abstract stuff. Not to be rude but its people like you who are responsible for poor education in the world, too many professors teaching difficult abstract topics first before showing the real world use of them, which is why videos like this exist.
You have to think of it in an "abstract way" too (or abstract algebra way) because in a space greater than R3 it's actually imposible to visualize it.
I have just learned about vector both in physics and computer science but no one could explain as clear and understandable as you. I love this playlist "Essence of linear algebra" and every video you make. Hope you will get more subscribers, I will share this channel to my friends.
This man singlle handly carrys every 1. Year Physics Student during pandemic through their exams. Thank you a lot!
Let’s be honest: y’all watch this for the dialogue
“I know this already”
“Ah but young padawan, all knowing, you are not. A subtlety, there is.”
I usually watch for the dialogue, but this time I actually watched for the educational content. This video made so many things much clearer for me.
I am 40 yrs old learning this for the first time....this brings a sense of completeness and tranquil to my life.....even if so in tiny bits. You should be proud of your service to mankind (and kids kind !!)
6:35 that flat sheet visualization really hammered home the span concept in 3D space
Many beautiful words were said about how helpful your videos are.
I'll just say, this is one of the first times when I'm watching an "educational" video without having my eyes wandering to the "recommened" videos.
You sir are doing a better job fighting ADHD than any pills I've tried!
Wow, I didn't expect the videos in this series to come out so fast.
Well, he did say for the first 5 vidoes one every day.
Oh... I guess I missed that.
He said it : 5 videos in 5 days, then one every 1 or 2 weeks :D
i think he does them all before posting them (i would do that to reduce pressure and more quality)
I'd love to see an 'essence of geometry' series
I would like some basic euclidean 2d geometry with some problems added in.
Euclidean geometry? Algebraic geometry? Differential geometry? Non-archimedean geometry?
I prefer abstract algebra hahaha
I studied linear algebra in one of my college courses. Towards the end I did manage to figure out what all these things mean and how to apply them, but holy shit why can't we just receive an intuitive explanation first and then go into the math.
every video, I try to write each valuable Information to craft it in my mind, I end with writing each possible word you spoke.
At the time I was at grad school, there was not a CZcams channel like this. You have no idea of the valuable public service you are doing with these videos. Thank you a lot and congradulations on the great job!!
You opened my 3rd eye, enormous respect and love for you teacher.
I am a computer science student and never once in class the professor told us how and why we use vectors in CS. So it used to be something I dread cause of the lengthy and complex definitions they give. You are a life saver sir. I can't describe how much this helps me.
This is so much better than what most books and lecturers do: throwing a rigorous definition at you, and then working on examples with the definition that we barely understand, hoping that we can learn as they do the examples
I really hope those lecturers can put down their pride and learn from this video. Or even just show this 10 minute video to their students. Saves students’ brain cells and saves their office hours
I have never let my schooling interfere with my education.
~Mark Twain (1835 - 1910)
Omg words of wisdom 😂👏
I've watched this series out of interest about a year ago. I did not see how helpful this would turn out to be! (Taking Linear Algebra 1)
THANK YOU, thank you, a thousand times thank you.
Dor Peled me to im talking linear algebra right now and I have to show a proof on the relation of the range being a subspace of the span and I had to go back to these vids to get an understanding
"Imma take a break from studying linear algebra and watch some youtube. . .nevermind"
This man is carrying me in my college Linear Algebra course.
This is an absolute pleasure to watch. While taking linear algebra, I had to form most of these mental models on my own-glad to see them illustrated so well!
I'm currently taking a linear algebra course in college and was trying to visualize how vectors act in 3 space, especially when you say it's dependent or independent. The book doesn't help and the instructor can only do so much, but I'm glad I found your videos. You do an amazing job and I'll keep watching as I learn the new topics to better visualize, and I will definitely recommend this to my classmates
finally, after graduation, i got intrest in mathematics👀😃😃
same with me broh
Who else here has completed a linear algebra course but still comes here for the joy of seeing 3b1b explain things?
I do! I'm here for the explanation of what a linear transformation actually *is*, because for all that we covered the theory in our course, we never actually saw what they looked like, so when they were pulled out for sketching conics I was a little thrown.
9:02 "If each vector adds another dimension to the span, they are linearly independent"---LIGHT BULB moment for me. Uni professors/GTAs couldn't explain this simple thing. Hats off to you sir!
These videos are absolutely incredible-you’re spoiling us!
Also, someday, for the coders among us, you must show us your workflow in detail! Please?
This is definitely the question I get asked most. Perhaps one day I'll make a video on it, but the workflow itself changes. Once I'm finally at steady state maybe...
You know a steady state will never happen! You can't get nothing from something! Schrodinger. That's why there are three blue and one brown!
What language do you use?
It is python, his GitHub is github.com/3b1b
Great. Thanks for sharing.
Man.. more than 20 years after leaving school, watching a video in a foreign language, I understand more than I ever did in school in my native language. You are a genius.
This has definitely opened my eyes, I love how you explain how linear algebra is being used and how you demonstrate each aspect of it. This definitely going to help me throughout my class.
the number of "eureka" moments in your videos is incredible ^^ these videos are brillant
This is the best way to learn linear algebra. Thanks for uploading this series. Wish there were some videos for multivariate calculus on your channel.
Sorry for the late comment, 3b1b has the multivariate calculus covered in Khan Academy.
You can't imagine how much this was helpful fo me!
I'm a chemistry student and I'm studying linear algebra and through your beautifully made videos with these really convincing animations you have helped me a lot.
I was just studying the maths behind vectors, matrices, span, basis etc.. without something visual I could refer to.
Thank you 3blue1brown, I think we are all really happy to have you for free on CZcams going through all this work to help us understand better.
We love you man!
As a person in architecture + design, I am so grateful for these videos. I missed math after so many years of not needing it, and wanted to teach myself linear algebra. These videos show that thoughtful design can enhance everything that we do, especially learning. Imagine if every class we ever took taught concepts like this... The movement, simplicity, and clarity made the topic not only easy to understand for a visual learner, but also soooo fun to learn. THANK YOU
Pure gold! Thanks for these!
I never fully visualized everything I've learned in math through high school and college until now, your videos are amazing and saving me for my linear algebra class!
I wish I had you as my Linear Algebra teacher. You explain so clearly in 10 minutes a concept which I did not get the entire semester, despite being able to mechanically solve more difficult math problems.
The way you know somebody is good at teaching their subject is when it can make a newfound appreciation and love in their students for the subject. I really love math when I see it through your lens, and I'm so glad I got to be able to have you share this perspective.
You just took us to another dimension. thanks for your efforts.
pure joy who needs netflix!? perfect! 20 years later after my studying but it's what I should have watched back then!
An entire week of fuzzy Coursera content was explained in a crystal clear way in a single video wow.
Got a test in linear algebra this friday and u literally just went over basically everything on it 10x better than my professor.
Because basis vectors are linearly independent, does that mean basis vectors always have to be at right angles to each other?
Good question, if two vectors are "linearly independent", it doesn't mean they are at a right angles to each other, it just means that they *don't* lie on a single line. So two vectors at a 5 degree angle from each other, for example, would still be a basis for 2d space.
frozenbagel16 No, it just means they are not collinear. When you talk about 3D space, it means they are non-coplanar.
I had to go so down to find this question, pin this comment, will be helpful to others.
frozenbagel16 I totally agree with this question I had the same thoughts in the video but awesome question
The answer to that question is not, but you can manipulate a basis and change the vectors so they are perpendicular one to another. This is called orthogonalization.
These videos are absolutely amazing. They're making the geometry of linear algebra not only clear but beautiful.
If someday CZcams shuts down, I hope someone will archive these videos. They're too good and should be seen by everyone
You're opening me a world. Definitive explanation of this subject, nothing more needed. Humanity should gratefully thank you!
the quote from Angus K. Rodgers in the beginning of the video make me cry. it was just beautiful
7:54 really helped me visualize that concept
My teachers never taught me the geometric concepts/idea behind all these terms. Thankyou very much.
Your end slide captures the essence of what makes your videos (3Blue1Brown) and another CZcams channel, QuantumSense, so brilliant. You start off presenting the intuitive concepts, and then carry the viewer into the fuller picture. This is THE right way to teach.
Too many others begin with abstract, sterile definitions plucked from nowhere, proceed to prove abstract theorems, and only then give the student any sense of what all of this means. No wonder students find these subjects daunting.
The animations leave things waaaaaaaay more understandable
Thank you for incorporating images and animations in your videos. It is impossible for me to understand a linear algebra concept without me seeing it in a geometric form.
The imagery and verbal descriptions are so beautiful and smooth. I love the way you paint these topics
I'm a student from Vietnam, we study math rigorously throught tests and exams but I still can't learn such concepts as intuitively as watching your videos. Thank you, for your effort.
Took two semesters of linear algebra and honestly passed those classes by just memorizing the patterns in how to solve the problems. Never actually gleamed any knowledge on any of it which is a shame. Thanks for these vids.
For the first time I understand linear algebra intuitively. What a great tutorial.
This video is 8 years back from when I'm watching, but this is still absolutely class and useful for me. Thanks a lot for the quality content!
There are people capable of explaining complex things in a way that people educated in the field can learn and understand them.
Then there's this dude that explains crazy stuff and makes it look like it was as simple as a kid's LEGO worm.
That takes some awesomeness!
Next one's gonna be sweet, can't wait!!
I love that classical music. It adds even more joy to the joy of actually understanding the simplest Linear Algebra concepts :)))
Holy moly for a whole year I had to memorise this without understanding it until now wow! I had to sit there with my jaw dropped for a few minutes after you mentioned linearly dependent vectors 😭 It all makes so much sense now!!! Thank you for the clear and very intuitive explanation!
Amazing to see someone make math interesting
Amazing! This single video has taught me things I was unable to understand in 3 hr lectures of linear algebra class. Thanks a bunch!
P.s The quiz was very helpful in deepening my understanding.
Im really excited for the next generation’s discovery given theyll be entering an educational system far supreme with these interactive visuals and free content. Now, how to restructure our scientific publications, that’s another step that needs much work
This is a beautiful series. I have been studying linear algebra for the last couple of months, and while I did understand the theorems and the algebra itself, I really did not understand what it all means. Those geometric representations really gave me a better understanding of what I am actually doing. Thank you so much!
I've been watching this series in awe for at least 4 years, and now I'm finally far enough in my education that I am taking linear algebra and I feel like I'm only reviewing stuff because of you. thank you
This is amazing. Thank you so much, sincerely, from a student eager to learn but didn't know where and how to start at linear Algebra . Thank you
I wish I found such videos back when I was being stuffed with each concept the wrong way. My love for math would grow more and more and will finally become something that can benefit me in real life. I am 35 years old, do you guys think there still time to set back and understand them again this way so I could use them the way they meant to be used?
I have never commented something like this on an educational video, but this is insanely helpful. I've always been good at math, and after taking through calc 3, linear algebra is the first course where I REALLY need something like this to help me visualize it to understand it. Nothing like this is ever shown in the textbook or in class. I'd be stuck with mindless memorization without this! Thank you
my man you just eloquently summarised like 2 weeks of my uni work that I was really struggling with and I am in your debt