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

[@moldyluke]

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

[@ThinkPositive00]

Before 0% understood

[@jonathansimmons3344]

ThinkPositive00 middle 50% understood.

[@PhysicsMath]

Well

[@ChristianRoland7]

nice

[@neelparekh1759]

Are you a professor now?

[@user-co4rr8cy9v]

@@neelparekh1759 Are you a professor now?

[@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 😢

[@ms_nana]

Yes

[@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

[@ChristinaWahlquist-h5z]

Thank you! -Christina, 48 year old mother of 7 children finishing a Secondary Math Ed I started 30 years ago. :) Your solution was clear and organized. Awesome!

[@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

[@みかちゃん-k4r]

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.

[@4sky]

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

[@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 ☺️

[@certifiedwavy]

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

[@unnamed1992]

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

[@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

[@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!

[@ryansuber6253]

I understood more in this 25-minute video than in my lecture on the topic today........ Thank you for this video

[@theekags]

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

[@rohhanbhardwaj]

how was it?

[@cjames9001]

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

[@tejasghodkhande3381]

Very Nice explanation!

[@reypope19]

You're saving my linear algebra grade, THANKS!

[@Novice0825]

I assume you've graduated by now!

[@liliashaymuratova6729]

It's fantastic! So straight-forward. Thank you so much!

[@GbyP]

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

[@artindesign2565]

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

[@EwaldBE]

I have always heard good things about Khan Academy and it definitely checks out. This video explained a topic I have been struggling with clear as day.

[@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

[@benjaminjongepier6826]

I love you Khan Academy

[@oliverandamms738]

thank you !! i know im not alone when i say that! this cleared everything up so neatly

[@supersonic174]

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

[@BrainGainzOfficial]

Yup!

[@patrickneal9288]

this saved my life

[@devashishbhake3173]

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

[@RawwestHide]

khan is a god

[@drrojas]

KHAN ACADEMY in HD , aaawwww yea!!

[@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?

[@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

[@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

[@hrzbltnrd]

thank you so much, finally a video i can understand

[@shriram6123]

Very nicely explained

[@rajaabubakar4104]

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

[@vincelunceford]

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

[@shameerrishad4189]

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

[@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

[@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.

[@ArthurTaylor]

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

[@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.

[@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.

[@hasunsri]

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

[@Clodidi1]

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.

[@majed1911]

There is a separate video for that

[@ThePearReviews]

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

[@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..

[@jacobm7026]

Mind. Blown.

[@TDefton]

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

[@Asdun77]

You explained it very easy thank you, god bless you

[@arjunselvam7]

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

[@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?

[@SaeedRanjbar]

anaaaaaaaaaaaaaaaaaaaazing video ! Neat Clear , thanks !

[@oneinabillion654]

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

[@JeremyLeeTW]

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

[@dezebarrow3663]

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

[@kenikozo]

ITS MAGIC!!!

[@hansgodoy6434]

thank u very much

[@梁廷睿-t5k]

Great!

[@콘충이]

Thanks!

[@roelheirman8398]

You just saved my ass :)

[@pianoforte17xx48]

Your brain*

[@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...?

[@GaryTugan]

awesome vid

[@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

[@javierzanet]

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

[@PrinceFX]

AWESOMENESS !!!

[@akshitajohar16]

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

[@MohamedElsheikh22]

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

[@tugbamacit4224]

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

[@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.

[@alepov]

thanks

[@lemyul]

thanks sal sal

[@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

[@SouthernHadoken]

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

[@human.earthling]

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

[@vatcherc]

THANK YOU!!!!

[@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

[@imbsalstha]

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

[@iLoveTurtlesHaha]

1:34 "We don't know that these are linearly independent" ... yes we do, there are 4 vectors in R3, one of them will be redundant, therefore those vectors are linearly dependent. Also, after you row reduced, you just needed to see which columns had a pivot point then go back to the original matrix and take those columns and they are the basis vectors for Col(A). ... eg. there was a pivot position in the columns for x1 and x2, the basis vectors were eventually determined to be column 1 and column 2 of the original matrix. Make sure you understand what is going on in the video though, it's really important that you do.

[@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?

[@meghnadsaha2469]

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

[@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 !

[@bakhtiareng.6392]

SALL KHAN is proud of MUSLIMS

[@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?

[@Warrimonk]

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

[@VicfredSharikver]

nice

[@rituparnameshram9397]

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

[@bojanglessr3]

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

[@pianoforte17xx48]

*nullsapce*

[@yuanguolang5352]

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

[@unfragger]

I LLOVE YOU

[@cvpadre]

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

[@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

[@conner1832]

"Nullsapce" in the thumbnail :^)

[@Sythesia]

Null Sapce

[@realvideosrv1879]

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

[@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.

[@qotyop]

X3 is freeee

[@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.