Dimension of the column space or rank | Vectors and spaces | Linear Algebra | Khan Academy

"The rank of a matrix is the number of linearly independent column vectors that can be used to construct all of the other column vectors."
Perfect. Thank you so much!

These vidoes make linear algebra so much more clear. I am always lost trying to decipher my teachers lectures and the text, but then i watch these vids and it all makes so much sense! Thank you !

Thank you sooooo soooooo much, you are much better than a loooooot of profs in my university

you just taught me in less than 13 minutes, what my prof failed to teach us for the past 2 weeks.
Thanks alot to take time out of your daily life to help out students, especially for free.

At the time this video series came out, all people thought about was that it was great and that it explained things better than university professors, which allowed people to pass. In the future though, these videos will be considered as strong evidence for why in person university education is obsolete. These kinds of series will be the new basis vectors for education ;)

thank you so much Sal Khan, though you released this more than a decade ago, these videos are still as useful and relevant. a BIG THANK YOU!

You explain in 10 minutes what my teacher could not in two weeks. Thank you for saving my grade :)

I passed this class in college, and idk how did I passed it, but I came back for engineering grad school, and you have saved me!!!! This is supposed to be material I should've already known to understand more advanced topics :/

dim(null space)=number of non pivoted columns where as dim(column space)= number of pivoted columns.
very nicely explained thank you

I have a linear algebra exam tomorrow. It's hard and I don't have a good prof and my text book is written in such a way that you have to unscramble what they are trying to get at... The guy I was paying to tutor me was terrible. I don't know why I bothered with all of that when you make something that seems SO FOREIGN so like, normal and not that hard! Your videos make me like linear algebra! (kind of... that might be an exaggeration) but THANK YOU SO MUCH FOR SAVING STUDENTS EVERYWHERE!

I hope that you know that you are the reason i graduated!!!! haha at U of I to say the least you come in clutch and i can actually understand what your saying

that video will definitely help me in my final exam tomorrow. thank you!

I donate some to khan .. So that this incredible work will not stop in future..

Best math teacher of all time!!!!!

i love you, man...
this is the first time i really understood everything someone told me about this stuff

thanks for making linear algebra easy, i wish my lecturer could explain things as simply as you do

this video lecture help to learn to basis , rank and dimension for the subspace of column vector which i did not find a good one anywhere. thank you

To calculate Col A, do u use echolon form or reduced echolon form, watching some videos on it and every video has different method. Some are using reduced and some echolon only. So im getting confused which one is supposed be used?

K I watched some videos, and I think it's what I said, except you do not go back to the original un-reduced matrix to find the linearly independent rows that comprise the basis for the row space of A: you just use those rows you found in the reduced form. Nicely, once you find the row/column space, it is an easy task to find the column/row space, since the reduction exposes both in the matrix.

thanks a lot , that helped me a lot before my final !

thx..i've been reading about the rank and i don't understand at all... thx 2 u i understand it now

Best video on youtube for this topic

thanks for making "ranking" easy for me to understand,

You are simply charming.saved my quiz day. Thank you

if you, patrick jmt, and thenewboston, and other like that were to open a college....your education would be way more valuable than any other college because the kids coming out of your college would actually know stuff

amazing... my linear algebra final is next tuesday. I only wish I knew about this throughout the whole semester. I bet I ace it with this help.....

This is so crucial. I’m thankful

I really love your videos, they are awesome. You explain things much better than most of the professors in my university. Thank you very much.
Although, you made a small mistake at 4:45. You said minus one plus minus one is zero and minus two plus minus two is also zero. Well, the second minus one and the second minus two were actually plus one and plus two. And minus one plus minus one would be minus two, not zero. :)

11:30 for dimension, your welcome

in this lecture you said dimension =number of pivot columns in c(A) however in the previous video called dimension of null space and nullity you said dimension= number of non pivot columns in N(B) so which one is it? # of pivot or # non pivot columns? or does that change based on if its a null space V.S a column space? and what is the difference between the two?Thanks.

THANKS SAL YOU ARE LIFE SAVER

Thank you for your videos! Very helpful.
Also, the dimC(A) = Number of columns - caracteristic of (A) = 3
:)

Thank you so much! :D This was so helpful!

Thank you much! That was really helpful! :)

thank you mr Kahn. you are doing Gods work my good sir

Thank God for Khan Academy.

Great Video! Thank you!

thank you so much, your lecture is sooooooo great!

When you rref the column space (or span), don't you have to like turn the columns into rows first and THEN go ahead and do the substractions? (And then turn the rows back to columns)

You save my life , thank you so much .

I start to realize that Khan is much better than my prof.....

do you mind putting better tracability on your videos so we can know which one is the next one

what if two pivot columns are the same in reduced row echelon form? Is the rank then 2 if you have 3 pivot columns as two are linearly dependent?

THANKKKKKKKKKKKK YOU SO MUCH . PLEASE PUT MORE AND MORE VIDEOS. EXPLAIN A SLOWER AND AT THE END OF THE VIDEO SAY MISPLAY WHAT YOU DID . JUST LIKE WHAT YOU ASIDE AT THE END OF THIS ONE. MORE AND MORE VIDEOS PLEASE

Thank you.

u are da bossss! im gonna ace my test tmrw!

Perfect!! Thanks so much.

6:50 dont you mean column 4?

@g1tfisted My thoughts exactly. My linear algebra teacher gave us all a take-home test that we have to turn in in two days, and no one really understood his explanation of column space and nullspace. Thank you so much for your help, this will surely help me answer some questions on my test. :P

Thanx I find it helpful :-)

Does span of column space define the space of vector space?

Who else is watching this 10 mins before their exam?!?!!?

This guy sounds like WoodysGamertag. Great vid.

So can we directly calculate the rank of the matrix to find the dimension and the linear independent vectors ?

Dimension (Number of linearly independent vectors) of Column Space: Rank
Dimension (Number of linearly independent vectors) of Null Space: Nullity
And Rank + Nullity = Number of Columns in the matrix.

Thank you so so much!! xx

Sir, you are awesome!

so the dimension of the null space = # of pivot variables of the original matrix and the dimension of the column space = # of non pivot variables?

Good job !

thanks man

very helpful!

Hey Sal, thank you so much for all your videos. So, pivot columns are always linearly independent right?

what is the difference between the dim. of subspace and col. space ????

How to figure out which the pivot column

are dimension and rank basically the same thing?

Hello Khan academy, you have left me extremely confused since by the theorem of linear independence: If a set contains more vectors than there are entries in each vector than the set is linearly dependent. I must have misunderstood you so in other words: are you saying that the subset of A(a1, a2 , a4) is independent or that A (a1,a2,a3,a4,a5) is independent?

thank you !!!!!

thanx u r just amazing

THANK YOUU!!!!!!!

may I know which text book u follow please

thanks! and I wished my bald , boring professor take some notes :p

thx

THAAANKK YOUUUU !!!!

ur perfect

so i can say, dim(C(A)) is equal with rank(A) ?

two people teach linear algebra. Sal and the guy who disliked this video.lol

i love you khan

No, it doesn't matter. There is no "one right way" to rref a matrix, you can do whatever you want.

you are awesome

Omg i love you!

why is the 3rd column not a pivot entry

ah I should've skipped right to the end but the rest of the video wasn't too bad to watch either.

Column space is column rank?

1 person needs to work on their clicking accuracy

Why does he sound like Tom Hanks, lol?

Hm. Well this probably isn't helpful to you now, but I -think- you literally just take the ROW that the pivot point is in, rather than the column.

so .... rank is nullity?

I still don't understand what a column span is

I feel your pain!

@hilaryeeee I Pay $4500 as an international student........ But it's not better than this............

lol same apart from I have to pay even more :(

I really think you're on to something...if our fucked up govt. ever decides to relieve the poor of their education (koch brothers have tried) a system like the one you have started would definitely even the playing field.

@lebplayer004 khanacademy [dot] org

i love u
*no homo*
:)

i pay $4500 and my prof still sucks like never before!

Thank you so much for this!

Thank you so much!

thank you!

lol same apart from I have to pay even more :(

thanku so much!!!!