Basis of a subspace | Vectors and spaces | Linear Algebra | Khan Academy

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Understanding the definition of a basis of a subspace
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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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[@khanacademy]

Most everyday jobs really don't require you to know what these things are. If you enter any scientific/academic field, the ideas of basis and subspace show up a lot. I've also had to use these ideas when I worked at a hedge fund (when we discuss "factors" that can explain movement of a portfolio--the factors are the basis vectors and the universe of potential portfolio performances is the subspace that it should span)

[@TrainPlaneBoat]

so u can span portfolios :o

[@aviroxi]

@@TrainPlaneBoat *spam XD

[@meshachchacha6216]

​​@@TrainPlaneBoatyes

[@Deuterium52]

You are such a phenomenal teacher sal. You explain things more clearly than most college professors. Linear algebra is considered abstract to us undergrads, which it is, but you illustrate things so clearly. It reminds one that mathematics is rooted in logical reasoning and natural deduction. The way you explain things, everything is logical. So thank you very much for these videos.

[@GameIsMyOxygen]

Totally agree,he is one of the best teachers. Whenever i am confused about something,and if he has a video on it (which most of the time he has),by the end of the video it feels too easy. xD

[@Abdul_Rehman6162]

The best method of explanation. I cleared my Vector Space Concepts

[@fermidirac8904]

math is not logical what are you saying

[@elecengrmka]

I could not understand anything in 6 classes of linear algebra in the university but few hours watching your clips have made everything crystal clear. May Allah bless you!

[@danielgonzalezisaiev9643]

Dear Mr. Khanman, Thanks to you, I have realized that when I grow up I want to be LINEARLY INDEPENDENT!

[@RiarArt]

OMG i learned 10 times more in 19 min watching this video than in last 2 weeks of Linear Algebra classes at University xD

[@nikitakundu2444]

Couldn't agree more!!! I'm a 2nd time grad student and finally getting a grip on this topic!!

[@abhinavkumarsingh6319]

Thanks a ton from the bottom of my heart to make such amazing and outstanding videos.

[@jeniferjanet4809]

You the real MVP

[@MexterO123]

I have an exam on wednesday, and with your help. I am confident in answering whatever my teacher gives in the Exam. This is was the first math class I actually was sort of lost in, and you have shown me the light. From now on your my teacher for higher level math.

[@mikohayle]

No one has explained this concept better than you! Thank you for all of your videos!

[@initrunlevel0]

thanks mr. khan, i am ready for linear algebra quiz tomorrow :)

[@agentoranges]

incredible. I took linear algebra last year and failed it hard. This time around, your explanations constantly make lightbulbs pop up in my mind.

[@dhruvbhati1347]

Isn't it true that if we prove vectors v1 and v2 independent then it will automatically prove that they cover span R2, and vice versa? If so then we don't need to solve it 2 times

[@MaroLuva]

That is the clearest definition of a basis I have heard in ANY college course I've had (including a whole semester of topology after linear algebra)

[@keziah9798]

hi, i just wanna say thank you for the video, it helps me understand better than my professor and my textbook

[@zoomboy6676]

All I can say is THANK YOU!!!!!!!!! You are the best prof for explaining linear algebra. Period

[@jonathansweet2230]

Thank you for helping me figure things out!

[@rdzdrd0]

Khan saves the day again!

[@tuoyazhuo8063]

The video explains the subject matter very well. However, when solving for the vector equation @7:20 (or when determining if the equation has a trivial solution), I think it is better to use augmented matrix instead of plain equations...

[@tis_i_sana]

Just a note: at 06:55 when he checks if they span R2 and then if they are linearly independant; this part is a little excessive. We can achieve the same thing by only checking if they are linearly independant and if they are then they must span R2

[@yarashamali8061]

his voice

[@Cliff86]

You should've mentioned that the basis of R^n will always only contain n vectors

[@Sporky0000]

Thank you for mentioning this! I was wondering if the minimum number of vectors that forms a basis was 2, but now I see it depend on the dimension.

[@linleybaruch738]

I've got this in my finals tomorrow, and need to re-learn the whole topic because it's been a few years since I've last done it.. Thanks for this :)

[@3hz3zv4eey6v]

This is better than my professor, textbook, and classmates combined. Thank you.

[@hadisyed1]

I would gladly donate whatever amount you require to get the best tablet out there. So that you may continue to enrich our lives with your undeniable gift of teaching the most abstract and unwieldy concepts in the simplest manner.

[@denett01]

There is a linear algebra playlist with all the videos in order on his channel. This video is number 20.

[@nasifhossain3441]

The best video for understanding what the basis is!

[@Rovshenification]

I'm a lazy to research something about Basis or anything about that but I think if I will watch your all lessons I will be lazy as always because everything is about linear algebra is in your channel. This is enough for me to do not any research about anything. Just click your channel and enjoy :D Thanks a lot!

[@mdrakibhossain2433]

You did a great job man.Fully understand the concept just in a few minutes which i didn’t in the whole last few years.

[@lhyx1990]

Every computer science student who wanna make awesome 3d games should definitely check this out. Best Linear Algebra tutorial ever!!!

[@allthingsacoustic]

Thank you for this video!! You just clarified a topic for me that had me completely lost.

[@s4ms4r4]

Shame our world doesn't financially reward people who do such amazing work. Imagine, if he went to Goldmansachs making complex crazy market products he could make a lot more money!

[@evachao4890]

基(basis)定義: 1. 可張成 subspace S 的向量 2. 線性獨立 作用: 可以用 basis 中向量的線性組合表示 subspace 中的任意向量 (是生成次空間的最小向量集) PS: subspace 的 basis 不是唯一

[@dawagyel]

thank you so much... u have always been life saver for me.. u are really the best

[@ullasrathodiitk5041]

its really great...

[@DavidDouglasJr]

I find this video much more enjoyable at 1.5X. Its ironic that this video is so redundant!!!

[@JosephHarner]

That would be entirely to effective, logical, and cost-efficient for any university to consider implementing.

[@fuahuahuatime5196]

For real... I love how this guy teaches. Thank you so much.

[@sudupati]

Absolutely brilliant Sir. no more words to describe...

[@strwbrryshrtcke]

Thank you soooo much, this helped alot! You made it so easy to understand too

[@pubududarshana4883]

Thank You. You are great.

[@kristkir]

Great video, I feel like they should get rid of the useless lecturers and just play these videos in my algebra class.

[@denebgarza]

@khanacademy for work, or not for work. I'd still learn this. It's interesting.

[@GeekBoy03]

I wish there are links to previous and next video in a series

[@josephdamen620]

You the man Khan

[@KhoaNguyen-jn1vr]

thanks for your lecture a lot

[@greenapplefever]

These are great videos. Excellent job.

[@fspopshovit]

Thanks for these video's, they helped me a whole bunch! Too bad I found them so late, my exam is in 2 days! Keep up the good work though

[@elektronic]

saved my life man, thanks a bunch.

[@Cyberpunk644]

never went to linear algebra class. cause prof sucks so bad.. you sir are the best!!!!!!!!!!!

[@metalninjadragon]

These are awesome, especially for a grad student in controls (yes another one) who is suddently smacked in the face with algebra symbols he hasn't seen in 10 years. Why don't you mention the connection between linear independence and the determinent though?

[@hardtunesbigvibes]

thanks.great explaination

[@oluwatoba11]

"In a Basis, you have no redundancy!"

[@azrnzala]

great video to jog my memory :) thanks.

[@bobmuijs7401]

You're a fraeakin' genius. I don't think I'm going to school anymore.

[@mentld]

Thank you! This helps me so much :) The videos themselves are great but would it be possible for the episodes to be number-ordered or could quotes and such be linked to previous episodes? KZbin doesnt show videos in correct order.. Anyways thanks so much and sorry for my english :D

[@vrchacho]

@norwayte Computer graphics won't have gotten anywhere without this.

[@debasishkalita3962]

you are great sir, when you say about the last video,can you provide the link as well sir?

[@eda7210]

Thank you so much sir

[@Xlaxsauce]

your the man Sal

[@khalidteama2441]

Amazing.... thanks you very very much!!!!!

[@khalidteama2441]

Everything makes perfect sense now... spent all day trying to understand my lesson until I came across this. Cant thank you enough!!

[@HK-sw3vi]

I didn't give two shits about vectors when this was published, yet here I am

[@sickpee]

Given a set of linearly dependent vectors, how would I determine which of those vectors I can "kick out" - so to speak - to get a basis for the space spanned by that set?

[@FileMaker_Consultant]

Got a question. If you have a 3x4 matrix and after row-reducing there are 3 pivot columns, leaving behind one free variable. Will it be a basis for R3?

[@elenaorins5284]

Because I cannot find a video, is there a video on extending a basis? Thanks a lot for the video by the way! My college hates using matrices

[@Incrue]

Sal, you rock.There's a chance you make videos of electronics or mechatronics? Thank you very VERY much

[@ArchilonValdir]

Thanks for this! I really needed a refresher!

[@johnbo7433]

This is 10 times better than my college professor

[@crumcon]

you know alot about math Khan

[@mdib2001]

Everyone: His teaching is phenomenal. Me: He always takes good colors. ✌😎

[@ShanieMyrsTear]

Question on the basis of a subspace. I see in this video we are using Basis' to find Spans. However, I don't see in the video how to find the basis itself?

[@statushood]

great video!! thank you!

[@blockbusstar]

Bill Gates give this man another million dollars !!! The ART OF EDUCATION !!!

[@kyleblosser514]

@kingroy2377 basically, the vector is still in R^2 since it is a column vector with i and j components, thus making it two dimensional. The vector would be in R^3 if it were, for example, [2 3 1], since that has three components [i j k], rather than two components [i j], or [1 0] in this example

[@aniketpanda9302]

Sir can you please make a video on finite basis and infinite basis with examples(especially polynomials).

[@Maibes]

this guy rocks my world

[@niceperson2]

Why not use a more general term-Basis of vector space rather than subspace. BTW very helpful😇

[@JetKMendoza]

Legend bro, keep up the awesome work! I think you should come teach at my university haha Definitely subscribed!

[@irombebekanandameetei925]

Definitely i love ur videos sir.

[@kamrannasir3871]

Is there no need for any third vector to be there to form a basis. Here we see that only two vectors are sufficient to form a basis.

[@catalinamaria9253]

This Mr just saved my butt

[@kanishka475]

Thanks a lot.

[@kingroy2377]

i am quite confuse about the basis, span, subspace..for example when you added another vector does it goes from R^2 to R^3, please can you explain ..thanks

[@veryfrostyjack3067]

amazing

[@MitzpatrickFitzsimmons]

isnt it then possible to define subsets of vector coords to infinity to and or of the the eleventy-seventh?

[@mengqi68]

My college education feels like a waste of money and time. Most professors are very unclear and expect you to know things that are not in the textbook or lectures.

[@fredhasopinions]

great video, but unfortunately none of these wonderful videos about subspaces and bases find how to actually FIND a basis of a given subspace?? I'm confused and distraught haha

[@kam1am]

This is awesome, I'd be screwed w/o this video

[@Chriscx]

WHEN YOU SAY LAST VIDEO CAN YOU PLEASE PROVIDE A LINK TO THE LAST VIDEO!

[@N1Kburchy]

If you go on their actual website it has them in order :) www.khanacademy.org/math/linear-algebra/vectors_and_spaces

[@jiaqiwu4512]

if you view the playlist it also has them in order

[@Babe_Chinwendum]

I think only one vector is enough to span R2, don't you think? That means the S you showed is not a basis for R2 using the same logic

[@harrietrj95]

Question: If the set S is not linearly independent, you can still have a span V of it but it is not a basis, but if the set S is linearly independent, then the span of that particular set S is a subspace of V and it is a basis?

[@MuffinsAPlenty]

That is correct.

[@Niharikajain72828]

Yes Yes what you said is correct and important for better understanding

[@raulmayers2072]

If the set S is not linearly independent meaning it is linearly dependent it can't be a basis for V but isn't it the span of S still a subspace of V?? It is just all the linear combinations of the vectors in V, so if V is a subspace wouldn't span(s) also have to be a subspace of V?

[@A-003]

I wish I have seen this one week ago, I have a linear exam in 36 minutes and i am sooooo gona get F*****. But thanks a million for that video

[@mwaleed2082]

So how did it go?

[@A-003]

@@mwaleed2082 well thanks for asking, I actually had a classmates that were very lazy, the professor had to make a curve down, i was not doing THAT bad in linear but I feared falling. long story short, I actually passed.

[@mwaleed2082]

@@A-003 well, 10 years ago I was in class 5 or 6 I guess. I didn't had any intuition on such math because we were already having a tough time understanding Algebra -1. I'm now in Mechanical engineering, freshman year. I wanted to go into computer science but my dad being an engineer himself wanted me to be one, he would have no other way. So I went into mechanical engineering and one day this thought popped up into my head, what CS fields can I integrate into my engineering? Therefore I started to search and came across Machine Learning. I started searching for pre-requisites on Machine Learning and apart from programming the most important thing everyone insisted upon was on Linear Algebra. Other universities in my Pakistan do teach Linear Algebra, but our university which is considered the Stanford of Pakistan, would replace Linear Algebra with Multivariable calculus's some concepts and Complex analysis. I figured I'm missing out big time! I already had a good intuition of matrices, echleon form, and therefore this qurantine I went on to study Linear Algebra. I've purchased Mike X Cohen's data handling and Linear Algebra courses via Udemy and I'm studying those and Khan academt's Linear algebra in parallel. I hope to finish these by the end of Quarantine. Hope you're somewhere good in your life now, I pray for your well being. Also I would like to know about your professional career, how have things worked out for you? I guess 10 years ago when you made that comment you were of my age and in freshman year as well, I'd love some advice and guidance.

[@A-003]

@@mwaleed2082 oh brother, I don't know what to tell you, the problem is back then we didn't have much development programs to advise us upon what is the most suitable thing to study for an individual, therefore my choice of studying (chemical engineering) was totally random, I was good in math and physics (sucked big time in chemistry) in IGCSE and AS level, by the process of elimination, I kinda found myself in chemical engineering. It is not totally bad, I didn't hate, but honestly I would rather study something that I would actually love (I honestly don't know what it is till now, 33 years old and don't know what is my true passion). Long story short I graduated . .. . found a job after 2 freaking years, in an O & G company, I worked in the field (it is practically slavery job bro, like physical Labour) they don't treat you as an engineer, some companies do that. anyways, situations lead me to work in the office for the past 2 years, I am still not treated as an engineer, I use only the skills of an engineer but not the knowledge I acquired in classes, honestly I feel all my education was for nothing. sorry for the long reply, please please please make sure what ever you are studying is what YOU and ONLY YOU wants, one of my wise friends once told me is that your passion is probably in the job that you enjoy doing even if they didn't pay you for it, (for me I really don't know what it is yet). I really wish you the best of luck brother

[@mwaleed2082]

@@A-003 oh man, this is exactly what I fear, except that I may not even get a job without 3.5+ GPA. Yes oil and gas industries are cruel. Especially ones like Schlumberger and Eni, ones that are far away from your home. Are you from Asia? I'm from Pakistan.

[@amanbhanse5645]

How to find basis of intersection of two subspace??

[@seen8]

i didnt even watch it buh i knoe its gonna be goodd..

[@owenarnst7729]

I love u Sal Khan

[@danieldings1]

OMG Finally!

[@norwayte]

... I mean.... in this case the general ideas are mathematically ... written down.

[@hebahatem3814]

يارايت يبقى فيه شرح بالعربى 😥😥

[@stokastia]

That makes 2 of us!

[@dibbidydoo4318]

I'm a high school student watching this and do not understand a single thing.

[@iLoveTurtlesHaha]

Start from the beginning of the playlist on the actual website. It's organized in a way to help you understand.

[@aneezabilal4503]

Well, I'm in my first year of A levels, and I have a further maths test tomorrow. It's already 2 am and I have to complete this by morning, even though I didn't understand a thing in the classes 😭

[@JammyTom]

If you are this far in math, you should know that you don't ask that question : P