Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra 

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

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

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

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

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?

ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Xcz-rVPvL2Y.html That's you when you meet him

@@Masardirasa i like people like you hhhhh

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!

@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

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 😂😂

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 😂 😉

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

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.

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.

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?

68th like ...nice

I agree

@@Gravitraxer_AangCZ 169th like …nice

Thats what I was saying right now! Exactly this!!!!

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.

@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?

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.

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 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 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 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.

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

Omg words of wisdom 😂👏

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.

Sorry for the late comment, 3b1b has the multivariate calculus covered in Khan Academy.

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

same with me broh

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.