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

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

Рет қаралды 810,308

15 жыл бұрын

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: kzfaq.info_...

Пікірлер: 185

@ThinkPositive00 10 жыл бұрын

Before 0% understood

@jonathansimmons3344 6 жыл бұрын

ThinkPositive00 middle 50% understood.

@PhysicsMath 4 жыл бұрын

Well

@ChristianRoland7 4 жыл бұрын

nice

@neelparekh1759 2 жыл бұрын

Are you a professor now?

@moldyluke 7 жыл бұрын

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

@CodSock 7 жыл бұрын

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 5 жыл бұрын

@colesmith5522 Жыл бұрын

@anindochoudhury1199 8 ай бұрын

yes :)

@pawaninfo7007 3 ай бұрын

Tomorrow 😢

@ThinkPositive00 10 жыл бұрын

After 100% understood

@sam2026 4 жыл бұрын

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

@kozukioden2406 4 жыл бұрын

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

@hugoirwanto9905 4 жыл бұрын

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

@shawnjames3242 4 жыл бұрын

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

@vishakamohan5336 3 жыл бұрын

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

@dripminic 5 жыл бұрын

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

@BrainGainzOfficial 5 жыл бұрын

True!

@ozzyfromspace 4 жыл бұрын

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

@fodaseodinheiro 4 жыл бұрын

check this kzfaq.info/get/bejne/btVldtCX1cvTZps.html

@alinac5512 3 жыл бұрын

Ty! Thats what I was thinking

@weirdcuriosity3690 3 жыл бұрын

Yeah

@ozzyfromspace 4 жыл бұрын

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 3 жыл бұрын

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.

@JeremyLeeTW 7 жыл бұрын

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

@Lolsashalol 5 жыл бұрын

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

@marekjanik9962 3 жыл бұрын

COVID-19: Oh no you won't!

@lozy497 2 жыл бұрын

same

@4sky 13 жыл бұрын

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

@Asdun77 4 жыл бұрын

You explained it very easy thank you, god bless you

@artindesign2565 2 жыл бұрын

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

@certifiedwavy 5 ай бұрын

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

@unnamed1992 12 жыл бұрын

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

@martinmarmo 8 жыл бұрын

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

@benjaminjongepier6826 9 жыл бұрын

I love you Khan Academy

@khanacademy 15 жыл бұрын

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

@deryakarabulut7805 4 жыл бұрын

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!

@hasunsri 11 жыл бұрын

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

@tejasghodkhande3381 3 жыл бұрын

Very Nice explanation!

@elohimlouis5677 11 жыл бұрын

I wish you explained every single subject math and computer related

@metallvaleeee 4 ай бұрын

thank you so much, finally a video i can understand

@shriram6123 2 жыл бұрын

Very nicely explained

@SaeedRanjbar 9 жыл бұрын

anaaaaaaaaaaaaaaaaaaaazing video ! Neat Clear , thanks !

@jacobm7026 5 жыл бұрын

Mind. Blown.

@drrojas 12 жыл бұрын

KHAN ACADEMY in HD , aaawwww yea!!

@PrinceFX 14 жыл бұрын

AWESOMENESS !!!

@tugbamacit4224 6 жыл бұрын

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

@GbyP 5 ай бұрын

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

@reypope19 13 жыл бұрын

You're saving my linear algebra grade, THANKS!

@Novice0825 4 жыл бұрын

I assume you've graduated by now!

@rajaabubakar4104 6 жыл бұрын

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

@supersonic174 6 жыл бұрын

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

@BrainGainzOfficial 5 жыл бұрын

Yup!

@cjames9001 14 жыл бұрын

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

@user-gb6qw4yv7p 3 жыл бұрын

Great!

@andreashaugstvedt8076 5 жыл бұрын

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?

@shameerrishad4189 2 жыл бұрын

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

@vatcherc 12 жыл бұрын

THANK YOU!!!!

@TDefton 4 жыл бұрын

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

@kenikozo 12 жыл бұрын

ITS MAGIC!!!

@theekags 3 ай бұрын

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

@rohhanbhardwaj Ай бұрын

how was it?

@user-bz8nm6eb6g 3 жыл бұрын

Thanks!

@iczyg 11 жыл бұрын

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 2 жыл бұрын

awesome vid

@patrickneal9288 2 жыл бұрын

this saved my life

@oneinabillion654 5 жыл бұрын

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

@MrCalhoun556 14 жыл бұрын

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 11 ай бұрын

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

@vincelunceford 11 жыл бұрын

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

@devashishbhake3173 3 жыл бұрын

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

@hansgodoy6434 4 жыл бұрын

thank u very much

@RawwestHide 8 жыл бұрын

khan is a god

@rkishei 12 жыл бұрын

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.

@roelheirman8398 9 жыл бұрын

You just saved my ass :)

@pianoforte17xx48 3 жыл бұрын

Your brain*

@alepov 11 жыл бұрын

thanks

@ccuuttww 6 жыл бұрын

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 5 жыл бұрын

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

@cvpadre 11 жыл бұрын

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

@vishnus2567 4 жыл бұрын

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

@Lucas-zd8hl 4 жыл бұрын

Yes or else we can't use it

@tejasghodkhande3381 3 жыл бұрын

yes

@arjunselvam7 Жыл бұрын

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

@SouthernHadoken 6 жыл бұрын

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

@yuanguolang5352 8 жыл бұрын

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

@akshitajohar16 2 жыл бұрын

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

@VicfredSharikver 15 жыл бұрын

nice

@NotmyYTchannel 15 жыл бұрын

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

@certified_vg2200 3 жыл бұрын

how old are you now?

@NotmyYTchannel 3 жыл бұрын

@@certified_vg2200 12

@NotmyYTchannel 3 жыл бұрын

@@certified_vg2200 jk 30

@bunstie5208 2 жыл бұрын

@@NotmyYTchannel wow still active 8)

@NotmyYTchannel 2 жыл бұрын

@@bunstie5208 yup og

@Liaomiao 11 жыл бұрын

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 3 жыл бұрын

are pivot variables always the linearly independent ones- Yes

@lemyul 5 жыл бұрын

thanks sal sal

@unfragger 14 жыл бұрын

I LLOVE YOU

@dezebarrow3663 2 жыл бұрын

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

@Europemaster 14 жыл бұрын

@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.

@ThePearReviews 11 жыл бұрын

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

@aaad1100 8 жыл бұрын

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 8 жыл бұрын

+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.

@Warrimonk 14 жыл бұрын

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

@javierzanet 14 жыл бұрын

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

@meghnadsaha2469 10 жыл бұрын

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

@MohamedElsheikh22 11 жыл бұрын

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

@linkmaster959 5 жыл бұрын

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

@BrainGainzOfficial 5 жыл бұрын

Yup!

@atharvajadhav2319 4 жыл бұрын

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

@DjVsal 11 жыл бұрын

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

@DrinkedTooMuch Жыл бұрын

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?

@TBV121 12 жыл бұрын

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 11 жыл бұрын

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..

@realvideosrv1879 3 жыл бұрын

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

@human.earthling 12 жыл бұрын

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

@bojanglessr3 10 жыл бұрын

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

@manpreetsaggi786 11 жыл бұрын

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

@orpheuspericles9582 7 жыл бұрын

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

@vishalgoel6690 6 жыл бұрын

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 6 жыл бұрын

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 6 жыл бұрын

The next video explains and visualizes that point. Thanks !

@test42356y 6 жыл бұрын

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

@BrainGainzOfficial 5 жыл бұрын

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

@conner1832 3 жыл бұрын

"Nullsapce" in the thumbnail :^)

@Sythesia 6 жыл бұрын

Null Sapce

@joodmu2002 Жыл бұрын

I love u

@slottedaloha649 5 жыл бұрын

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

@qotyop 11 жыл бұрын

X3 is freeee

@manpreetsaggi786 11 жыл бұрын

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

@pianoforte17xx48 3 жыл бұрын

*nullsapce*

@rituparnameshram9397 3 жыл бұрын

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

@Knot2goodAtIt 9 жыл бұрын

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 9 жыл бұрын

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.

@louaialfaori7978 11 жыл бұрын

a Gizzillion Times agreed!!!!!

@amdperacha 11 жыл бұрын

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.

@dickson710 10 жыл бұрын

一個字：無得輸

@teomazzaferro7040 11 жыл бұрын

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 6 жыл бұрын

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

@jojogaroot 8 жыл бұрын

there's a mistake when you row reduced the matrix

@StirsMYCookiez 8 жыл бұрын

+Abdulmajeed Garoot ?

@ericroncin436 8 жыл бұрын

If you're talking about the result of row 2 in the first step, he did the calculations and then multiplied the row by -1 to make his leading one positive. He just never said it.

@AgueroIsKing 7 жыл бұрын

No he didn't. You can multiply rows and columns by scalars, it doesn't change anything.

@abdullahkardas8887 5 жыл бұрын

thanks for your comment