Introduction to linear independence | Vectors and spaces | Linear Algebra | Khan Academy

[@RJA10001]

you've helped me in grade 8 math, which was 6 years ago. Here I am in 2nd year university, and you're still helping me, but now I'm getting a math minor. =D

[@godson200]

You just reminded me of when I was in 8.

[@SokumenTop]

Same here 5 years after you

[@AshokKumar-ix4nf]

@@SokumenTop now i will comment in 2026 😂

[@naman4067]

I will comment on 2027

[@aykuttosun4607]

Greeting 🙂How was it? Are you happy yo get a math minor after 6 years later?

[@RobynHode8]

Sal Khan is invaluable to mankind.

[@Knot2goodAtIt]

This guy is such a beast.... so many people would have failed their classes without him lol

[@khanacademy]

It is usually introduced in 10th or 11th grade in Algebra II or Pre-calculus.

[@Sahil_Ombale]

Oh no

[@vonStalhein2]

Albert Einstein said(in relation to what genius means), "It should be possible to explain the laws of physics to a barmaid." Sal is a living example of that.

[@xxemilyannexx21]

You are a godsend! I have spent all year learning about matrices and eignevalues and linear algebra and partial derivatives and not understood. i have spent 3 days watching your videos and it is clear now! thank you soooo much!

[@Ohmsip]

I have a linear algebra midterm tomorrow, you're my saviour sir.

[@eman5300]

Sal my bro you are the king never met you but you one of my dearest mentors 💯 helped me through education and both life 😭😂 keep this dream alive once I graduate imma try to join your team and keep your approach to education alive

[@zmaska]

THANKS KHAN!!! IN 6 minutes you made it clear for me waht SPAN means!!!! My teached wasn't able to do so neither the book!!! YOURE A GENIOUS

[@pleaseenteraname1215]

It's been 4 years but i think you should see 3blue1brown's video he has great math content he also used to work at khan academy

[@nizarch22]

@@pleaseenteraname1215 lol he probably graduated college and now has kids, but sure 3blue1brown is great.

[@skaai]

you are awesome sir! for those of us who are up studying at 3:30am, you are a godsend! not to mention you should rename your name from khanacademy to linearalgebrafordummies! keep breaking it down for us non-mathematics dummies! plenty of us like math, but don't speak "math" natively...

[@uclocnguyenvo422]

his voice is addictive :) god level of inspiration

[@yeonhyungjun674]

You are like life saver of my intellectual journey. I got into physics 3 months ago, and I learn all sorts of calculus with you and khan academy, and now I am learning linear algebra. first I tried to study this new subject with books, then I got frustrated cus I didn't understand any of explanation and proof in the book. but now after watching this lecture, what's in the book started to make sense!!!

[@iamabean]

Khan , Thank you so much for all the videos. After watching your videos,I just realize how stupid I was when I spent my time reading the god damn linear algebra textbook. Your explanation is very concise but very informative

[@KabooM1067]

Math textbooks are the worst at teaching math I swear to god I. Never benefited from them except the exercises. ._.

[@twonulator]

James Stewart Calculus and Saxon Algebra series are the only books worth a damn. The rest can be committed to the flames.

[@vivienne_lavida]

Yes, James Stewart is the best calculus book out there.

[@saviourmukuka2833]

Kkkkkk just like me 😝😂

[@ProTawN]

This guy can explain all of this so much better than my Professor.... thank you so much!

[@Peter_1986]

+ProTawN *Typical professor:* "Today we are going to introduce New Concept. :-|" "This New Concept is defined by the general formula written on the board. :-|" "Okay, so here is one example. :-|" *Khan:* "Hello. 8-) ♪" "Let's talk about New Concept in this video." "So let's say I had... I don't like this color... oh well, anyway, it will do the job. 8-)".

[@MohaMMaDiN55]

Where ever I go to a scientific youtube channel, I will never find someone who is more satisfying than you. Your channel is the best according to my experience of watching so many scientific vids on youtube.

[@mohammedishfaqmostain4835]

I can’t believe I am still talking help from this guy even in uni

[@fitnesschannel9070]

Sal, you're a God's gift. And that's not an exaggeration. Seriously. How is your existence possible...

[@skkamalshah]

Half of the semester gone, I learned absolutely nothing. today, before 2nd mid linear algebra started making sense to me from ur videos. Thanx man!! Take a bow!!!

[@iLoveTurtlesHaha]

OMG, I'm in the same boat as you. I hope I don't fail L.Alg. :(

[@mohinikaur9463]

I really prefer long videos where an in-depth analysis of the topic is done. 15-20 minutes is good to take. Thanks for great stuff!

[@prominence1351]

"I want to get back into habit of making shorter videos" Next video is 2 minutes longer

[@Hercules003]

My uni professor and GTA failed to teach me this simple thing- I thought I was stupid! In 15 minutes, I have realised I am not stupid.

[@wingbaby28]

Thank you for making these great videos!! I am taking linear algebra, and the way my professor explains those concepts is very hard to understand. I feel way better after watching your videos.

[@BARaaz04]

Hey Sal, you are fantastic. You have god gifted teaching style.

[@davin7151]

An alternative way of showing that any two non parallel vectors a, and b have a span of the whole cart. plane is to first pick an arbitrary point denoted x,y, then draw a vector b * inf through that point, where 'inf' = infinity. I.e. infinitely long line through the point. Then you can show that a * inf passing through origin will always intersect with b * inf passing through the point, since non parallel lines always meet.

[@RangeK1D]

I am in Uni now, I see why people come here, your saving my Linear Midterm today :)

[@FilipePauloMax]

Thank you very much...something that was very abstract for me, now no matter what exercise you give me, i can THINK right and solve it not with formulas but with your explanations :) thanks a lot you should be paid for that!

[@7angels844]

as long as the only linear combonation for v1 and v2 that equals 0 is when c1 and c2 is 0 then v1 and v2 are linearly independant and span V

[@ReneUlloa74kg]

I love you. I am in grad school at NYU for econ and you are better than the book.

[@Waranle]

Thank you, Thank youuuuuuuuuuuuu Sal, To start again posting these Linear Algebra. All i needed.... great.

[@studysmartderps7305]

Returning student and forgot a huge chunk of math even basic algebra😅. But still able to keep up because of youtube straight to point explanation.

[@aadityagupta6838]

O god! U listened to me 😇Being Maths hons. Graduation student i was really struggling with this algebra n these videos of yours turned out to be miracle for me.. Thanx khan academy, u really cleared my concepts...

[@42muslimah]

aww, thanks for noticing Sal! I had noticed that videos were getting pretty long, but they're still super helpful, so I didn't wanna say anything. :)

[@ChatGPT-]

Bro explained this all when I just started going to school 😭❤️

[@peters.9371]

the terminology was confusing me, until you explained it properly, thanks!

[@郑入]

linearly dependent means that one of the vectors in the set can be represented by some combination of the other vector in the set

[@MrWaqasyaqoob]

you a very good teacher , i like your teaching style very much . You are doing great work , thanks

[@Xlaxsauce]

you make much more sense then my professors.

[@onlyAerik]

I especially appreciate some of this vector-centric teaching, and especially the matrices. I just got to a point in one of my electricity textbooks where I'm defining instantaneous values of waveforms as vectors, and they went through it fast and weird.

[@wajyt312]

thanks! its much more understandable when u draw it out. unlike my lecturer...

[@excalibercuberdavid4681]

Im watching this in 2021. I was 7 when you created this video.

[@micahtonning7967]

Thank you for posting.

[@alexgomez4525]

Thank you man, you have made me understand this, much better than my teacher

[@imyafatha999]

"i don't wanna do it that thick" - salman khan

[@lynsikeye3714]

Question: Why is it relevant if they are linearly independent or not? Take the first example, even if they were independent, because there isn't a z-component of the vector, it will always only be on the R2 span.

[@MrMereum]

becomes relevant later on for finding null space basis i think

[@dfsfklsj]

Mathematics is the same anywhere in the world... It's not like languages that change dependent on country.

[@KydroxHD]

0:04 乇乂ㄒ尺卂 ㄒ卄丨匚匚

[@jaytaffer9641]

My gf asks, "Is that a linearly independent vector or are you just happy to see me?"

[@guscosta1]

you make much more sense than my professor and I dont even speak english

[@kheo46]

brilliant video, laid out the fundamentals well

[@ruggerobenetti8770]

These videos are dope. Thank you so much

[@relentlessfella]

Thanks dude... this is the best explanation of the concept so far and it was very much comprehensible =)

[@meinankitv4766]

Give this guy my college tution fee..

[@fahimally3923]

YOU GOING TO HEAVEN AND I'LL MEET YOU THERE. THANKS

[@MrDannyX3]

Thank you so much!

[@humbatgasimov6452]

Wonderful video.

[@MegaBugBugBug]

Yeah that had me confused at the start as well. Go to his first video where he tells that this is just a way of representing data he could be doing it like [0,2] or like (0,2) even if its a column it is just basically the same thing. If you take [0,2] and turn it on top so that 0 is at the top it's all good. I say stick with how your teacher does it they will probably prefer it in tests. I think he just does it like this to prepare people for matrix which are later on.

[@extraterrestrial16]

Should there have been '+' and '-' signs for the R set notation to specify the directionality of the vectors (vector space) ?? Given that the two vectors were either in the positive or negative planes..

[@ahmedaltalib5981]

Tnx, it is now easier.

[@sarvanray3971]

Very to the power infinity nice video.. really u r teaching style is unique as well as best. U teach this using its practical significane and imagination and exactly is my way of learning anything and everything

[@astrophy1586]

Those who have put a thumb down are really buffoons.. No one can explain simpler than this man😍

[@Zerpentile93]

You explain it very well :).

[@Peter_1986]

@gommaq Hus You mean "redundant", right? It means that something doesn't have any meaningful purpose, and that you can safely ignore it. For example, you only need two 2×1 vectors to specify a point in ℝ², so any extra vectors will thus be redundant, since they are not necessary at all.

[@maven12LA]

Sal khan is GOAT

[@thetruereality2]

Does this mean that any two set of vectors in a two dimensional plane can span that plane? 5:26 -6:20 Thank you for an amazing explaination

[@joyzhou4917]

Actually, no. If a pair of vectors in a two dimensional plane are linearly dependent, then they cannot span that plane.

[@mangathayarumaddukuri2709]

Great explanation

[@liverpoool4lyf]

woow, i'm studying it now for my Computer Science degree...Thumbs up if you are to!

[@qualquan]

Just re-discovery of head to tail vector combination using bunch of novel terms. Old wine in new bottles.

[@rnibir2769]

5:00

[@luayabd]

thank you very much

[@jishuenkam]

@bernicedan well we do have some matrices in moderm math

[@TheJuga]

Thanks man the Berkeley textbooks are sort of convoluted

[@taekwonwelchify]

DUDE.... YOUR THE SHIT. YOU JUST GOT THE NATURAL TEACHING SKILLS... LOL FOR REALS.

[@Vivekanand44]

Very useful

[@kayrstar8965]

Thank u

[@thenekoling]

thank you!

[@estidavezi5825]

Thank you it turned out to be very helpful :D

[@Razorxyzify]

+estida vezi you single gurl?

[@Judobrinez]

she is mine! jk

[@Judobrinez]

nice bruh

[@divvelaharsha826]

good explanation

[@quidquopro1185]

The guy that do thease tutorials, does he also do Dark Souls walkthrough? The voice is identical!

[@PhilNailedIt]

Awesome video, better than 'sneezing panda.'

[@shengzhou6170]

Wait if the matrices written in orange are linearly independent how is it that they are able to still represent anything in R2? so what's the difference between linearly independent and dependent

[@humbatgasimov6452]

Good.

[@girlonlaptop]

thanks!

[@Gommaq]

Thank you :)

[@nidhishgautam9043]

Sal is the saviour of 99%. Humanity. Except for those 1% >140iq

[@caseclosed2519]

hey look there's a pivot column in every column when it's linearly independent. we're learning woo

[@arpithams]

what abt vectors[7,2] and [2,3] are they linearly independent..

[@dimitrisntaganos5879]

yea

[@tangoforthemango]

this is awesome

[@SahibzadaInstitute]

Beat of luck my dear sir >>>>>

[@Jack-cm5ch]

6:40

[@hemaniprincy5226]

Hi sir. This is the first time I am watching your tutorials. They are quite interesting. But I have a doubt in the span of linearly dependent and independent set Linearly dependent vectors spans lie over a straight line. And linearly independent spans R2 r8..... However when we considered the set of three vectors where 1 is a linear combination of the other two.....they are linearly dependent r8.... Then how come their span is R2. The concept is clear.....but I am getting confused in identifying the linear dependent and independent vector.....

[@sherjr7406]

you're amazing.

[@deephknallareddy6670]

span the splane

[@highestintheroom-podcast741]

when you said "those three guys" you remembered me 3blue1brow..

[@trollzeus8288]

how can the whole R2 plane be represented by a linear combination of just 2 vectors? the 2 vectors are essentially 2 lines no? Then how can every point in the 2 dimensional plane be represented by just 2 lines?

[@dormoshe4237]

You can add a part of the vector to another part of the other vector, for example, if you have (0,1) and (1,0), by adding them n times you reach (n,n). Notice that n stands for any real number. Also, you can add one of them n times and them other m times. This way you can reach any point on the plane

[@trollzeus8288]

Oh so, the 2 lines can be "scalarly multiplied" to represent any point in a plane?

[@dormoshe4237]

Exactly. This is not always the case, try watching the linear dependent and independent videos

[@trollzeus8288]

Dor Moshe perfect. Thank you!

[@brucelee8318]

👍

[@tedtdu]

What if R^5? so they can't be expressed by X,Y Coordinated graph? Will you please shed some light please.

[@mlaumusic]

You're awesome

[@miekzand]

Amazing video's! You already helped me out with Biology, Chemistry and now with Math. Thanks :)

[@AkoPanotti]

anyone seen buffalo 66? "Spaaaning time together, just spaaning time"

[@karannchew2534]

What does R 2 (or R squared) represent? Could someone enligthen please?

[@godson200]

Xy axis, or in general a plane

[@jaytaffer9641]

dependent, independent... meh I'd rather call them 'linearly extraneous' and 'linearly ?something?' makes more sense than saying dependent to me.