Introduction to linear independence | Vectors and spaces | Linear Algebra

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

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15 жыл бұрын

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Introduction to linear dependence and independence
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Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
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Пікірлер: 149

@RJA10001 7 жыл бұрын

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 3 жыл бұрын

You just reminded me of when I was in 8.

@SokumenTop 2 жыл бұрын

Same here 5 years after you

@AshokKumar-ix4nf 2 жыл бұрын

@@SokumenTop now i will comment in 2026 😂

@naman4067 2 жыл бұрын

I will comment on 2027

@aykuttosun4607 Жыл бұрын

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

@RobynHode8 10 жыл бұрын

Sal Khan is invaluable to mankind.

@Knot2goodAtIt 9 жыл бұрын

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

@vonStalhein2 8 жыл бұрын

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.

@khanacademy 15 жыл бұрын

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

@inkofficial3571 Жыл бұрын

Oh no

@MohaMMaDiN55 5 жыл бұрын

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.

@eman5300 3 жыл бұрын

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

@xxemilyannexx21 12 жыл бұрын

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!

@mohinikaur9463 8 жыл бұрын

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!

@yeonhyungjun674 2 жыл бұрын

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!!!

@wingbaby28 7 жыл бұрын

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.

@ProTawN 9 жыл бұрын

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

@Peter_1986 8 жыл бұрын

+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-)".

@iamabean 9 жыл бұрын

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 8 жыл бұрын

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

@twonulator 7 жыл бұрын

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

@vivienne_lavida 7 жыл бұрын

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

@saviourmukuka2833 5 жыл бұрын

Kkkkkk just like me 😝😂

@KingIDMF 12 жыл бұрын

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

@uclocnguyenvo422 5 жыл бұрын

his voice is addictive :) god level of inspiration

@aadityagupta6838 2 жыл бұрын

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...

@mohammedishfaqmostain4835 2 жыл бұрын

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

@onlyAerik 14 жыл бұрын

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.

@Waranle 15 жыл бұрын

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

@FilipePauloMax 14 жыл бұрын

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!

@skaai 14 жыл бұрын

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...

@skkamalshah 6 жыл бұрын

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 6 жыл бұрын

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

@kheo46 3 жыл бұрын

brilliant video, laid out the fundamentals well

@fitnesschannel9070 2 жыл бұрын

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

@gazmaska7593 9 жыл бұрын

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 5 жыл бұрын

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 3 жыл бұрын

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

@Hercules003 4 жыл бұрын

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.

@BARaaz04 6 жыл бұрын

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

@alexgomez4525 7 жыл бұрын

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

@relentlessfella 12 жыл бұрын

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

@ruggerobenetti8770 5 жыл бұрын

These videos are dope. Thank you so much

@sarvanray3971 3 жыл бұрын

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

@MrWaqasyaqoob 14 жыл бұрын

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

@peters.9371 4 жыл бұрын

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

@micahtonning7967 5 ай бұрын

Thank you for posting.

@wajyt312 11 жыл бұрын

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

@Zerpentile93 11 жыл бұрын

You explain it very well :).

@davin7151 7 жыл бұрын

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.

@ReneUlloa74kg 6 жыл бұрын

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

@prominence1351 8 жыл бұрын

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

@studysmartderps7305 Жыл бұрын

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.

@RangeK1D 11 жыл бұрын

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

@extraterrestrial16 9 жыл бұрын

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..

@asif7240 Жыл бұрын

Thank you!

@mangathayarumaddukuri2709 4 жыл бұрын

Great explanation

@MrDannyX3 5 жыл бұрын

Thank you so much!

@thenekoling 13 жыл бұрын

thank you!

@ahmedaltalib5981 11 жыл бұрын

Tnx, it is now easier.

@Vivekanand44 5 жыл бұрын

Very useful

@imyafatha999 9 жыл бұрын

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

@42muslimah 12 жыл бұрын

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. :)

@Peter_1986 8 жыл бұрын

@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.

@Xlaxsauce 14 жыл бұрын

you make much more sense then my professors.

@ChatGPT- Жыл бұрын

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

@tangoforthemango 13 жыл бұрын

this is awesome

@sherjr7406 11 жыл бұрын

you're amazing.

@luayabd 14 жыл бұрын

thank you very much

@girlonlaptop 13 жыл бұрын

thanks!

@7angels844 24 күн бұрын

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

@user-ko5il5bo6c 2 жыл бұрын

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

@hemaniprincy5226 6 жыл бұрын

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.....

@divvelaharsha826 5 жыл бұрын

good explanation

@excalibercuberdavid4681 3 жыл бұрын

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

@Gommaq 14 жыл бұрын

Thank you :)

@TheJuga 13 жыл бұрын

Thanks man the Berkeley textbooks are sort of convoluted

@miekzand 10 жыл бұрын

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

@mlaumusic 7 жыл бұрын

You're awesome

@guscosta1 13 жыл бұрын

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

@jishuenkam 14 жыл бұрын

@bernicedan well we do have some matrices in moderm math

@KydroxHD 6 жыл бұрын

0:04 乇乂ㄒ尺卂ㄒ卄丨匚匚

@SahibzadaInstitute 7 жыл бұрын

Beat of luck my dear sir >>>>>

@kayrstar8965 3 жыл бұрын

Thank u

@astrophy1586 3 жыл бұрын

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

@quidquopro1185 8 жыл бұрын

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

@tedtdu 8 жыл бұрын

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

@shengzhou6170 5 жыл бұрын

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

@MegaBugBugBug 11 жыл бұрын

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.

@lynsikeye3714 5 жыл бұрын

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 5 жыл бұрын

becomes relevant later on for finding null space basis i think

@KodiBrehdon 11 жыл бұрын

anybody know the software, that are used for sketching (black background, etc)? thank you

@meinankitv4766 4 жыл бұрын

Give this guy my college tution fee..

@qualquan 2 жыл бұрын

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

@fahimally3923 3 жыл бұрын

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

@maven12LA 3 жыл бұрын

Sal khan is GOAT

@EugenioMejia 13 жыл бұрын

what program do you use?

@thetruereality2 8 жыл бұрын

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 8 жыл бұрын

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

@PhilNailedIt 12 жыл бұрын

Awesome video, better than 'sneezing panda.'

@estidavezi5825 8 жыл бұрын

Thank you it turned out to be very helpful :D

@Razorxyzify 8 жыл бұрын

+estida vezi you single gurl?

@Judobrinez 8 жыл бұрын

she is mine! jk

@Judobrinez 8 жыл бұрын

nice bruh

@caseclosed2519 7 ай бұрын

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

@liverpoool4lyf 12 жыл бұрын

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

@deephknallareddy6670 8 жыл бұрын

span the splane

@jaytaffer9641 3 жыл бұрын

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

@taekwonwelchify 12 жыл бұрын

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

@dfsfklsj 11 жыл бұрын

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

@arpithams 13 жыл бұрын

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

@dimitrisntaganos5879 4 жыл бұрын

yea

@Jack-cm5ch 6 жыл бұрын

6:40

@Gommaq 14 жыл бұрын

what does redandent mean?

@rnibir2769 Жыл бұрын

5:00

@nidhishgautam9043 9 жыл бұрын

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

@trollzeus8288 7 жыл бұрын

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 7 жыл бұрын

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 7 жыл бұрын

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

@dormoshe4237 7 жыл бұрын

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

@trollzeus8288 7 жыл бұрын

Dor Moshe perfect. Thank you!

@brucelee8318 6 жыл бұрын

👍