Null space and column space basis | Vectors and spaces | Linear Algebra | Khan Academy

Figuring out the null space and a basis of a column space for a matrix
Watch the next lesson: www.khanacademy.org/math/line...
Missed the previous lesson?
www.khanacademy.org/math/line...
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.
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to KhanAcademy’s Linear Algebra channel:: / channel
Subscribe to KhanAcademy: kzbin.info_...

[@moldyluke]

You explained in 25 minutes what I have been confused about for the past 200 minutes of my class. Amazing

[@CodSock]

Anybody else have their linear algebra exam coming up too? haha you saved me once again khan academy, very clear and easy to follow.

[@zoomboy6676]

Me

[@colesmith5522]

me

[@anindochoudhury1199]

yes :)

[@pawaninfo7007]

Tomorrow 😢

[@ThinkPositive00]

Before 0% understood

[@jonathansimmons3344]

ThinkPositive00 middle 50% understood.

[@PhysicsMath]

Well

[@ChristianRoland7]

nice

[@neelparekh1759]

Are you a professor now?

[@ThinkPositive00]

After 100% understood

[@sam2026]

ThinkPositive00 lol, you have two separate comments. old KZbin was something else

[@kozukioden2406]

I love how both comments have the same exact number of likes ! Math students are so precise lmao

[@hugoirwanto9905]

@@kozukioden2406 wow 5 months later and its still have the same number of likes

[@shawnjames3242]

@@hugoirwanto9905 It still has the same number of likes 223. How far will it go? I am curious...

[@vishakamohan5336]

@@shawnjames3242 Yes. It's 259 on both now

[@dripminic]

A faster way to find the basis for the column space is to rref and then take the column vectors with pivots

[@BrainGainzOfficial]

True!

[@ozzyfromspace]

dom you're right! I noticed it too and had an aha! moment. Life of a math junkie lol

[@fodaseodinheiro]

check this kzbin.info/www/bejne/bqCYdKCcpcqibMk

[@alinac5512]

Ty! Thats what I was thinking

[@weirdcuriosity3690]

Yeah

[@ozzyfromspace]

The number of pivot variables = number of independent basis vectors that make up the column space of A. Very insightful, Sal! It took me a while to process but now I get it ☺️

[@user-ii5li7zj6k]

I'm just gonna say again, I don't really understand what my professor said but I'm able to understand the explanation from this video. It really helped me a lot, no matter I'm gonna fail this subject or not, thank you for making this video.

[@Lolsashalol]

i've got a feeling that i'll get my bachelors in Mech Engineering with this channel

[@marekjanik9962]

COVID-19: Oh no you won't!

[@lozy497]

same

[@JeremyLeeTW]

great for the review of basis, null space and column space for a matrix !

[@artindesign2565]

Ohhhhhhhh thankxxxx a lot....!! Finally I understand the difference of null and column space and it works for creating basis.

[@Asdun77]

You explained it very easy thank you, god bless you

[@unnamed1992]

OMG YOUR A GENIUS. I CAN'T BELIEVE I LEARNED THAT.

[@certifiedwavy]

thanks, i do not why i could not understand this but your video did the trick!

[@benjaminjongepier6826]

I love you Khan Academy

[@martinmarmo]

Very enlightening video! One question though. What software do you write on? I'd love to take notes in class using the same method

[@reypope19]

You're saving my linear algebra grade, THANKS!

[@Novice0825]

I assume you've graduated by now!

[@4sky]

2am in morning..."ill let you go for now" "yes!! im free! i can go to sleep!"

[@tugbamacit4224]

adamsın adam!! (trying to get it for a day long. finally you made it. thanks in advance.)

[@tejasghodkhande3381]

Very Nice explanation!

[@mayahna]

thank you so much i have a final in 4 hours and this made everything simpler

[@elohimlouis5677]

I wish you explained every single subject math and computer related

[@SaeedRanjbar]

anaaaaaaaaaaaaaaaaaaaazing video ! Neat Clear , thanks !

[@cjames9001]

this 25 minute lecture puts 3 weeks of lecture in class to shame, very helpful

[@PrinceFX]

AWESOMENESS !!!

[@metallvaleeee]

thank you so much, finally a video i can understand

[@hasunsri]

most probalably....self study...........or.........one good teacher(lecture) who knows the subject deeply....not by just passing the exams.....by feeling maths....

[@MrCalhoun556]

I think it makes a bit more sense to apply Elementary Row Operations upon the Matrix before figuring out the Column Space. You'll see already before if the system of equations collapses the vector to a line, plane or 3d hyper-plane. It also has then a nicer form to check for the results of the Rank-nullity theorem.

[@ArthurTaylor]

So when the determinent is zero, the system of equations collapses down to a line?

[@supersonic174]

so if there are free variables in the reduced row echelon form, does that mean that it is linearly dependent

[@BrainGainzOfficial]

Yup!

[@jacobm7026]

Mind. Blown.

[@drrojas]

KHAN ACADEMY in HD , aaawwww yea!!

[@shriram6123]

Very nicely explained

[@rajaabubakar4104]

this video should be of maximum 5 mins....but u are awesome in extending videos

[@vatcherc]

THANK YOU!!!!

[@TDefton]

So in order for the column space to be Liniarly independent, the rref would have to be the identity matrix, right?

[@user-gb6qw4yv7p]

Great!

[@GbyP]

This man has saved so many people's grades, about to take my linear algebra midterm rn 😅

[@shameerrishad4189]

I have a query: are pivot variables aka dependent variables & free variables aka independent variables?

[@kenikozo]

ITS MAGIC!!!

[@andreashaugstvedt8076]

What happens if you have a column consisting of only 0's, regarding the null space basis? Wouldn't that mean that the respective x-variable is neglectable?

[@iczyg]

Can a vector be in both a the Null space AND the Column space of some set of vectors? Or is it one or the other...?

[@khanacademy]

That's not exactly giving me the best incentive to finish

[@deryakarabulut7805]

Hello, is there not a mistake done in the first place when you were subtracting 2 times row 1 from row 2? You said so but you subtracted row 2 from 2 times row 1 and it changed all the result. I try to understand linear algebra and everything coming up with it so I may be wrong but this is opposite to what I learned from MIT open courseware and what you said in this very video. Please clarify this point for me or I ill get lost!

[@user-bz8nm6eb6g]

Thanks!

[@GaryTugan]

awesome vid

[@devashishbhake3173]

this video is pretttttyyyyyyy old yet very relevant in 2021......

[@patrickneal9288]

this saved my life

[@roelheirman8398]

You just saved my ass :)

[@pianoforte17xx48]

Your brain*

[@hansgodoy6434]

thank u very much

[@vishnus2567]

When we do the echelon reduction, do we need to make sure that the pivot elements need to be 1?

[@Lucas-zd8hl]

Yes or else we can't use it

[@tejasghodkhande3381]

yes

[@RawwestHide]

khan is a god

[@oneinabillion654]

Took me 1 day to understand span subspace basis null space column space and then remembering it

[@vincelunceford]

yeah i totally agree... but he tries to prove it more theoretically

[@ccuuttww]

the last part may not necessary to find the basis u can just pick it form the reduced encholen form which have pivot in each column in this case it is column 1 and 2

[@zoomboy6676]

But he just proved that columns 1 and 2 are sufficient for finding the basis

[@cvpadre]

Thanks for the video. Hope you keep up the good work, which obviously you are =0)

[@rkishei]

I wouldn't say it's so much over-explanation rather than thinking out loud. At least for me, this helps, not because I don't know how to subtract (subtraction being one of many things he 'over-explains'), but because I can keep track of every assumption he's making.

[@NotmyYTchannel]

OMG... I was just on this studying this topic right now... and you posted this up like 10 minutes ago... WOW!!

[@certified_vg2200]

how old are you now?

[@NotmyYTchannel]

@@certified_vg2200 12

[@NotmyYTchannel]

@@certified_vg2200 jk 30

[@bunstie5208]

@@NotmyYTchannel wow still active 8)

[@NotmyYTchannel]

@@bunstie5208 yup og

[@alepov]

thanks

[@Liaomiao]

are pivot variables always the linearly independent ones? can't you write the pivot variables in terms of the free variables here as well? ack it's kinda coming together for me... thx khan

[@aryankumarprasad1574]

are pivot variables always the linearly independent ones- Yes

[@SouthernHadoken]

there easier way to figure out the basis. it is the original columns that correspond to the pivot columns in its RREF.

[@VicfredSharikver]

nice

[@arjunselvam7]

This is the single most redundant way to explain that pivot variables determine the column space but I finally got it

[@lemyul]

thanks sal sal

[@unfragger]

I LLOVE YOU

[@dezebarrow3663]

Even though i finished this video, i play it back just to hear his voice :'(

[@ThePearReviews]

Its easier to say that the pivot columns of A form a basis for Col(A) :P

[@meghnadsaha2469]

YEA IT IS MOST IMPORTANT FPR EVERYONED , BY THIS WAY I THIK ANYBODY CAN LEARN MATH S BIN SIMPLE WAY

[@linkmaster959]

Can the basis of the column span be the columns with pivots in rref?

[@BrainGainzOfficial]

Yup!

[@atharvajadhav2319]

But why did he referred pivots from original one but not from rref?

[@aaad1100]

Curious, when you first proved that X3 & X4 were "free" variables, is that enough evidence to consider those vectors redundant and exclude them from the final linear independent set, or was that just coincidence?

[@faisaladel5034]

+aaad1100 It's even more than that ,seeing that in the reduced echelon form that the non zero rows are just 2 ,and the number of columns (variables) is 4 ,then you should figure out there is two additional variables or additional redundant vectors.

[@akshitajohar16]

Where are next videos , please tell can't find them

[@yuanguolang5352]

any one could help me to find the basis of left nullspace?

[@MohamedElsheikh22]

The basis of Nul(A) is the same spanning set of Nul (A)... I think you forget to say that!

[@javierzanet]

Well because you have 4 vectors in R3 so you can tell that they are linearly dependent.

[@Europemaster]

@khanacademy he is probably being sarcastic or just a throll, you are doing amazing job with your amazing explanations, dont let that anonymous idiots make you lose strength to carry on. Have a nice day.

[@Warrimonk]

Very helpful thanks, too bad I find it impossible to stay away in any sort of linear algebra lesson *yawn*

[@TBV121]

I think you made a mistake on your second computation. -2 x Row 1 added to the remainder of the entries in Row 2 should give -1, 2 and 1, not 1, -2 and -1.

[@manpreetsaggi786]

It's not you, it's just the human nature that can't accept the truth and the truth is majority of the teachers here don't care if the student learns or not.(not all cuz I have some great Profs at my school). But most teachers here just work for their pay check. That doesn't happen in India. People care more about each other. Now this guy explaining everything for free, that's the kind of spirit we need in teachers her. I don't want them to teach for free but just care more than they do..

[@DjVsal]

thanks again ! well ,i'm gonna forget mine LA teacher but not you.

[@orpheuspericles9582]

shouldn't the no. of basis vectors be equal to the dimension of the subspace??

[@vishalgoel6690]

Orpheus Pericles No, because here you can see that he put 0 for x3 while proving that v4 is redundant and put 0 for x4 while proving that v3 is redundant. So, we can get rid of both v3 and v4. Also, the basis of a subspace need not span all the points in the graph because the span of the subspace can be limited. For example, here, the span is limited to a plane in R^3. What we can say is that the number of vectors in basis need not be greater than the order of dimension.

[@ahmeddesoky8434]

For the point you mentioned @Vishal Goel, " the basis of a subspace need not span all the points in the graph ".... I think it is not as per the definition Sal gave in a previous video that the basis is the minimum set of vectors that spans the subspace ! Also, till now I am not totally convinced how the number basis vectors of a subspace to be less than the subspace order !?

[@ahmeddesoky8434]

The next video explains and visualizes that point. Thanks !

[@manpreetsaggi786]

Dear friend he is talking about the education standards of the US which are very very low as compared to other countries. What you are given in 12 grade her, I was given that stuff in 9th in India

[@lukeharrison1655]

Is column space the same as the image of the matrix?

[@BrainGainzOfficial]

The set of all images. Usually referred to as the range!

[@human.earthling]

haha, at 0:06 ...CURL over... ..really INTEGRATE everything...

[@DrinkedTooMuch]

So we have weird exercises to do as homework (tho we havent even done ANY exercises on this topic, all they did was throw empty definitions at us and expect us to be geniuses) where it says "Which vectors(b1,b2,b3) are in the column space of A?" A= 1 1 1 1 2 4 2 4 8 And thats all the info we have. How does one solve it?

[@bojanglessr3]

to moeb32, he said he was doing 2r1-r2 not r2-2r1...

[@rituparnameshram9397]

who are those ultra genius 93 people who disliked this video?

[@realvideosrv1879]

At the end, didn't he mean to say column space of A "C(A)" ? Instead of column span of A?

[@joodmu2002]

I love u

[@pianoforte17xx48]

*nullsapce*

[@conner1832]

"Nullsapce" in the thumbnail :^)

[@Sythesia]

Null Sapce

[@qotyop]

X3 is freeee

[@slottedaloha649]

About getting RREF, you've made a mistake (that actually not criticall, but anyway), when you subtracted 2 times row 1 from row 2 you said the one thing and did another one, you didn't subtract 2xR1 from R2 but added 2xR1 to -R2

[@louaialfaori7978]

a Gizzillion Times agreed!!!!!

[@amdperacha]

Right... that's exactly what i meant. And, I'm very curious what your source is for that statistic, cuz I sure as hell find it questionable how you came to that conclusion. PS I'm not american, so I could care less even if you wanted to offend them.

[@Knot2goodAtIt]

I feel like he never missteps, but this was definitely the harder way to find the column space...why not just out it in a matrix and get the leading ones? Maybe that's what you did, but it definitely seemed more consuming. I had to stop watching the video before I got confused...

[@OmegaCraftable]

There wasn't a clear goal that he was trying to get to. He wasn't doing all these steps just to get to the final goal of the linearly independent set of vectors spanning the column space of A. You need to interpret this video as being more of a exploration in the the relationships between a matrix, it's null space and it's column space, rather than an explicit problem solving exercise.

[@dickson710]

一個字：無得輸

[@manpreetsaggi786]

An average kid here need a calculator, an equation sheet for an exam and it's provide, where as any of that stuff in Indian schools is strictly prohibited. I am not talking about the small schools in the poor villages. I am talking about the prestigious schools which we have many

[@ArthurTaylor]

How did I pass this subject? This is so confusing 😭

[@Hemraj_basnet]

still confused

[@teomazzaferro7040]

just because people are in linear algebra doesnt mean they can follow simple calculations, there's some people in my class that are really dumb

[@sakhatbanda1529]

if you speed this up to 1.5 , it essentially feels like a man trying to win an argument against a whamen

[@claudiavidican]

This doesn't actually teach you what a null space is.. this basically teaches you some trick to figure out the basis of a subspace.... waste of 20 mins.