Null space 2: Calculating the null space of a matrix | Linear Algebra | Khan Academy

[@jughead18sp]

8:30 ... should be 3 X4 rather than 3 X2. great video :D really helpful

[@abdallababikir9154]

he fixed it already

[@LeinysandMassey4life]

more helpful than a 50 minute lecture

[@AliSinaAmini-r9c]

50 hours you mean

[@Crazy1793]

besides for a errors @5:27 and @5:55, these is very helpful! much better than my 2 hour lectures

[@tauceti8341]

10:52 Span of Nullspace - the vectors broken-up in addition Form: N(A) = x_1 [v1..] + x_2 [v2..] N(A) = ((RREF(A))

[@PoeyJerkins]

These videos are life savers...

[@nighttfallNZ]

Thanks mate, you're a legend. Easy to understand and follow! Keep up the good work!

[@Kerendipitous]

"zero minus zero is zero!" he says it with such shock.

[@Rovshenification]

This is the Method 1. We can use Method 2 which is called Special Solution. We can apply this method 2 after rref(A) , then we must determine our pivots column and free variables. We have to take free variable as 1 and zero. (For example we have 2 f.v x1 and x2. firstly we must take x1=1 and x2=0 and calculate our combination, then x1=0 , x2=1 and again calculate). Thanks for this lesson :)

[@CrimsonCorn1]

My math lecturer just regurgitates the textbook into power points. This helps a great deal. Thanks!!!

[@Marvelocity93]

These videos are great, LA is quite abstract in lectures but now it makes sense :)......@ 7:21mins, shouldn't we have had -2x four instead of -1?

[@Marvelocity93]

It's fine, Sal noticed the error.

[@Nachtmusiks]

He makes everything so easy to understand. It's almost a waste to go to class.

[@Things4Free-Things4Free]

Thank you sooooo much, you are an amazing teacher and i would love if you be my teacher for all the courses i take. I see you better than a loooot of teachers with PHs.

[@sidaksingh90]

My Math professor who has a degree from University of Chicago couldn't explain as well as you can!!! I vote you for Professor of the year!!!

[@maynafiesta]

5:27 and 6:42 calculating mistakes there therefore except these, a very succesfully video

[@dzhao4n]

Mistakes happen, what this video aims for is to teach the concept of Null spaces. I personally think it has done a great job on that.

[@BabyXGlitz]

such an effective delivery of a material, magical

[@Brenstar]

Never been happier in my life to identify a span! Thanks Khan!

[@bencolleymusic]

It's okay Sal, you're not the only one whose mind wanders in Linear Algebra :D Great help as always.

[@imRyRy]

The second one: It's the set of all vectors 'x' for which Ax=0. Another way to write it out: N(A) = {x ∈ R^n | Ax = 0} "x is an element of the n dimensional space of real numbers, that satisfies Ax=0 for some given matrix A"

[@cmburnsie]

Great video, as always! Just a question: When you reduced the third row (with the second) to 0,1,2,3 - shouldn't it be 0,-1,-2,-3 ? In the end, it will be a 0,0,0,0-row anyway, but just to easy my mind - am I wrong? (it was about 5 years since I took a math class, so I'm probably wrong) /John, Sweden

[@ricardoV94]

It is the same, you can multiply the row by a scalar without altering the equation, so you can just multiply yours by -1 to get the same result. He just did the subtraction differently: 4*row1 - row3

[@Samtheweirdgeek]

Thanks a lot! this opened up my brain

[@Tobyonekenoby]

Thank you so much! I have had a hard time figuring out the null space and thanks to you i am now able to solve it :)

[@abubakarqasim5602]

JazakAllalh Khair for all that you've done, going for my exam now after Jummah for Linear Algebra.

[@SciHeartJourney]

I caught your minor mistake because you've trained us so well over the years. I had thought I was going crazy, but then you caught it yourself. 😄 Thank you for all of your excellent videos, like this one.

[@DisdainforPlebs]

I can see forever nao! Thanks so much man, the guy at our university sucks in clearly explaining stuff, this really helped!

[@rezahomam7454]

thank you that was a great video.

[@lolgayvids]

Omg this was so helpful! Better than my Prof!

[@ajayraikar3929]

This is really very helpful for collage student. Thank you khan academy

[@davidolatunji5110]

He isolates x1 and x2 because they are lead variables and writes their equivalent equations in terms of x3 and x4 because they are free variables, they can be anything.

[@wunboonail]

Sal, I run a small maths coaching college and you kick my butt as a teacher and also in maths

[@natisaver]

khan is literally carrying me from high school through college LOL

[@AllOutOfGum]

it is also a valid row operation to subtract 4 3 2 1 from 4 4 4 4 and put that into R3

[@brightforeigner]

oh my god. you may have just saved my linear algebra grade. Thank you!!!

[@surfing_youtube]

You made a blunder at 7:21. It should be -2x4 and not -x4.

[@everlastinGamer]

Sharad Keshari he fixes it.

[@giaotran6910]

oh my god after three weeks and one failed midterm, I finally got it. Thank you.

[@opmike343]

God, if I never have to do linear algebra again it'll be too soon.

[@triggerhappy899]

thank you so much for all your help, keep up the good work

[@0selth]

Just multiply it by -1 and it yields the same row.

[@AbuBishir]

Wow. He has over a 1000 videos now! When I first heard of him he only had like a hundred

[@opliik]

Thanks sal, you just helped me with my homework =)

[@RedBloodCat1]

14 years old and still worth it weight in gold

[@Liaomiao]

what is the null space? the entire Ax or just the x vector that when multoplied by A gives 0? i kinda get the rest but the definition of the null space itself.

[@Lolsashalol]

I wish my professors were as competent

[@azn1ici0us]

mann, life saver before exams

[@nomoreheroes93]

We all know Sal didn't really make any mistakes in this video, he was simply seeing if we were paying attention.

[@Timmy-zl1gu]

great teaching , I'am subsribed too ,and i am chinese so i spelled wrong:D

[@lol58697]

can't thank you enough! :)

[@sportsbeast80]

It doesn't matter because you can just scale that row by -1 and get 0 1 2 3

[@MuuMuusWorld]

You can multiply R3 by -1. In the end, it'll still be the same.

[@triplethreat789]

Do you to reduce the matrix to Reduced Row Echelon Form when possible? Like for example in an invertible square matrix?

[@MassiveZero]

@9:36 Why x3 is equal to 1 x3 and 0 x4? Where is he getting this from? Same question why x4 is equal to 0 x3 and 1 x4?

[@timetopwn3429]

He gets it from the vector where it says [x1 x2 x3 x4] . The answer is in your question. x3 is equal to x3, but not equal to x4, and the same for x4. Think about it in numerical terms. 2 = 2, but not 2 = 3. 3 = 3, but not 3 = 2

[@davidolatunji5110]

He isolates x1 and x2 because they are lead variables and writes their equivalent equations in terms of x3 and x4 because they are free variables, they can be anything.

[@MassiveZero]

@@timetopwn3429 Thank you

[@MassiveZero]

@@davidolatunji5110 Thank you

[@tron12100]

Thank you, great video!

[@basimismail2093]

you da boss munn!

[@Nachtmusiks]

Khan Academy should open an actual academy. I'd attend.

[@DookieBlaster]

Thank you

[@dfsfklsj]

It might be way too late, but I'm going to say it anyway... The operation is still valid because the signs don't matter when the sum of the polynomial is zero. -1, -2 and -3 are the exact same as 1, 2, and 3.

[@Car1ll]

THANK YOU!

[@TheAllisonBreen]

@KingHenryXI Other instructors who are jealous of Khan :) Flawless indeed. As many other comments say this guy puts my Lin Alg instructor to a pitiful shame. Thank you so much Khan you are saving my butt.

[@nikitakokal1]

THANK YOU!!!!!!

[@xoppa09]

We need Khan University.

[@lampzzzzz]

dude ur awesome..!

[@Thewerwolf]

You need to fix x2 to x4 in equation 2 in green.

[@nomoreheroes93]

-1 is only a scalar and you're multiplying a zero vector so the outcome is not changed.

[@joshmacdonald5999]

6:20 why did you touch the top row?

[@migbadrule]

TANK YOU THANK YOU THANK YOU!!!!!

[@RodrigoLopezF]

What technology do you use? Thank you!

[@JacobShepley]

at 7:35 he wrote '3x2' instead of '3x4'

[@Tadeleh]

good stuff

[@ytfu666]

thx aaaaaa lot~~ u save my quiz tmr lol WP

[@Agendasuicid]

So those two vectors are the basis for the null space right?

[@Tekttskskwdi]

can we say N(A) = N(rref A) = N(Row echelon form A) right ?

[@karelspinka3031]

What if I have the opposite problem? Let's say I've got several vectors forming the null space and I want to find a matrix where the matrix times any linear combination of null space is equal to the zero vector. Thanks

[@sillylittleboy93]

Luckily in this case it doesn't change the answer because you could just multiply by -1 to get 0 1 2 3 | 0

[@irmaksaglam3782]

thank you so much! and can you please move the mouse less the next time?

[@benfarrelly188]

thanks bossman

[@fauxpace]

He multiplied the 3rd with -1 and added 4 times the first row i think

[@ibrahimabdalla1175]

I don't think Khan academy has let me down once.

[@elsavelaz]

how did he know to use the purple equation to solve for x3 and the green one for x4? How come you can't just use the same equation for all of them? And is he solving for them because they are free variables, or because x1 and x2 were already solved, or does that all mean the same thing?

[@chunkypandya]

dude come and lecture all our maths courses lol! salman bhai bohot dumda

[@VinothKumar-qo7ry]

Superb

[@RsGhost1]

Yea I guess, still it would have been more clear if he'd said that he was multiplying by -1.

[@oneinabillion654]

AMEIIZING~

[@wyphonema4024]

It just like solving a homogeneous equation to me.

[@RsGhost1]

I think you made a mistake at 5:53 you are supposed to substract 4 4 4 4 from 4 3 2 1 so you should get 0 -1 -2 -3.

[@ineedaname004]

i think at 7.40 there is some mistake. x2+ 2x3+3x4 not x2

[@078matthijs]

@AbuBishir he has almost 3000 videos ;)

[@Buddy-qb8sw]

Why is the null space of rref equal to that of original matrix?

[@ny1fanta]

my prof does it using determinate

[@Phagocytosis]

That's not what he said he was going to do though... both ways work, I believe. I did it the way you said as well, but Sal got to the same answer.

[@CSharpGuides]

wow I love you.

[@doordie7560]

i couldn't have easy explanation then this

[@yustinayasin5539]

why the zero vector is in R3?

[@akshaysingh4712]

i dont understand why you had to reduce the number above a pivot variable to zero. we were thought it was only the number below a pivot variable that to be reduced to zero. This is what is said on my lecturer's notes???

[@jeoantapo]

***** Great explanation!

[@1994anmol]

I got 1,0,-3,2

[@PhenomDubstep]

So why do we have to learn all this complicated way of doing things? Why not just reduce A then say that the free variable columns written as vectors are the null space of A?

[@motovlog8746]

6.29 ....why the first raw changes

[@birdsal432]

you make a fair few mistakes but thanks for the help

[@ZHeisenberg]

aww, i didnot wait to 9:40...

[@BahumbugFlop]

Yeah but then just multiply by negative 1

[@cmburnsie]

Update: ...when you reduced the third row with the FIRST row