Basis, Dimension, Kernel and Image

[@GeyserFlicker]

Was worried when I saw this was 56:57 mins long, but honestly this is the best explanation I've come across. Better than my lecturer and Khan Academy. Keep up the vids!

[@paulb8264]

Thank you for the Extremely clear and slow paced explanation. My professor waffled about for on these 4 concepts for 2 weeks and I never could make heads or tails of her work. I hope your students acknowledge how lucky they are to have you.

[@eveexd8325]

with this video I understood what I couldn't 1 month now,thank you:)

[@sagorbiswas9442]

খুব ভালো লাগলো স্যার

[@jeo2222]

very helpful, it would be nice if you add a table to contents so people can skip ahead to a section they need help with

[@StellarArt]

I know this is an old vid, but you are such a life saver. I learned more from this vid than I learned from my professor, and it's really sad tbh. Thanks for this!

[@slcmathpc]

Glad you found it useful :-)

[@GHCrandall]

This really helped clear up the fuzziness. Before all I could do was find explicit calculations. Now I feel confident in the concept.

[@leao1985]

Very clear and concise explanation of these concepts. Thank you!

[@yenmeng]

Your handwriting is perfect wow.

[@lmk6006]

I am so grateful your explanation in details. it is so much helpful for preparing my final. i was confused many things when i learned it in classl. Thank you so much again.

[@aarongeno9708]

Well explained but you made a minor error when finding the elements in the kernel. The last vector should be t [0 0 -2 0 1],but you have t [ 0 0 -2 0 2] . Since x5 = t and not 2t.

[@slcmathpc]

Aaron Geno You are absolutely right! Thank you for pointing this out.

[@Sarah._.s889]

Very clear and well explained Thank you very much🙏

[@davidrolston6014]

Love this!!! So straight forward and easy to follow! Cannot thank you enough!!!

[@LwaziNzimande]

Damn, I didn't want to watch this at first, but I'm glad I did. Perfect explanation

[@zoranjovanov8564]

haha was watchin this video before exam just for a recall xd its makes me laugh the way u describe everything. in a good way of course :) . comparing to my teacher u make it just so easy and smooth . we the people need this . big thanks man ,

[@leylaalkan6630]

+Zoran Jovanov Totally agree with you.

[@HolyDevill]

Thank you. Your explanation helped a lot and allowed me to take the leap to understand other videos and other parts of the algebra course

[@C.G.W]

you make me so sleepy when I m watching this!

[@hankzhou279]

Thank you so much! Your video makes more sense than my prof's lecture.

[@flawns]

44:18 for vector 3, variable 5. it's positive 1t not positive 2t. Unless I'm mistaken

[@slcmathpc]

Correct!

[@flawns]

such a simple mistake and yet we all do it, every now and then. Thanks for the videos

[@ms_1918]

best video on these concepts. Thanks

[@poojitha]

Perfect. Life-saving.

[@slcmathpc]

I very much like the idea of saving a life. ;-)

[@von6693]

I'm confused. I thought if there were degrees of free then the vectors were not linearly independent?

[@ahmetcanaytekin6744]

45:21, V3 is not correct. It should be [0 0 2 0 -1] I guess.

[@MrCheryeater]

You rock! This is the best explanation, I've heard or seen.

[@MacAndSwiss]

Incredibly helpful. Thank you.

[@siphomazibuko4864]

thank you so much, I was not understanding it at all, but now ....

[@mvas1l]

45:27 shouldnt the last element of v3 be 1 since you didnt scale it?

[@yanji3026]

Very helpful and clear! Thank you and wish high mark in exam.

[@LwaziNzimande]

This video is perfect for dumb people like me. Haha, just kidding, but it's such an amazing video. Explanation - on point. Pace - On point. Examples - On point. Everything - On point.

[@dionysismaniatakosmusic]

Really helpful video,thank you!

[@vaniagrawal4142]

Thank you so much for this video, it is so easy to understand!

[@user-pt-au-hg]

Thank you for helping me understand, much appreciated, Stephen. :)

[@nuculabs]

At 47:56, how do we put the vectors in a matrix [-3 41 0].. or [-3 2 0 0 0]

[@Sam-db8gv]

Thanks a lot man. Very well explained

[@mwanthidennoh1232]

very nice teaching ad elaboration in deed thax very much

[@karammm]

thank you! this helped me a lot!! text book is so confusing.

[@karammm]

btw I like that you talk slow. not sleepy at all!

[@chicharito1454]

ehhhhhh im stuck with Orthonormal Basis of image for a 4x4 matrix.... is it just following the steps to get the image and divide each of them by norm to get the orthonormal one?

[@slcmathpc]

The vectors in an orthonormal basis also have to be pairwise orthogonal. After you find a basis of the image of your matrix, you will need to apply the Gram-Schmidt Algorithm to transform your basis into an orthogonal one. Check out: kzbin.info/www/bejne/mnSxaK2wd6d4rbs All that's left after is to normalize the vectors by dividing them by their respective norms.

[@emmanuel3047]

Down to earth explanation..respect

[@von6693]

So the section around 22:00, the method he uses to find the basis of the plane is - I thought - the method for finding the basis of the null space. Is that not correct?

[@slcmathpc]

Both cases consist of finding solutions to a homogenous linear system.

[@victorialy3346]

had to watch it at speed 1.25 it is very lengthy, makes me uneasy even though really deep explanation thank you!

[@amrhamcho1853]

the tutorial is great, very helpful, thank you

[@jadenpieterse7056]

Thank you. Much appreciated

[@putinscat1208]

I followed everything that was done, but why is it done? The free variables confuse the hell out if me. I was under the impression that solving for n variables with less than n equations wasn't possible?

[@slcmathpc]

+Putins Cat Any homogeneous linear system with more variables than equations has infinitely many solutions. Consider the simple example x-y=0. The solutions to this equation are all the points on the line y=x.

[@putinscat1208]

slcmath@pc Sure, but what is with the r,s,t variables? I can replicate what was done, but why? When you multiple matrices, you take each row against a column vector, but then you can also take a matrix and split the columns into vectors and say cV1 + cV2 + cV3. I have no idea what that is about.. Is it just something that is what it is?

[@davidmburu9257]

Best explanation bro,,keep it up

[@user-mz7fh2hy6x]

I thought if there is any solution except 0,0,0 it is linearly dependent. here there is solution -6,1,0 why it is not linearly dependent? Also about dimension, I thought detention is 3 because there is 3 variables which are x,y,z

[@slcmathpc]

If the only solution is the zero solution, then the vectors are linearly independent. If there exists a nonzero solution, then the vectors are linearly dependent.

[@user-mz7fh2hy6x]

Yes.. there is nonzero solution which is -6,1,0 then it should be linearly dependent according to definition as you said, but you said that is linearly independent.

[@jiwoolee2470]

you sound like a white khan academy

[@slcmathpc]

Haha! I will take that as a compliment ;-)

[@kyhinds]

looool!

[@masamieiri4669]

if sal khan didn't use his name for his videos and site i would assume he is white too

[@sandrapaolacastrobeltran5655]

hellooo. How do I determine the free variables of the system? at the minute 40:37

[@slcmathpc]

Variables that do not have a leading one in their column of the reduced matrix. ;-)

[@sandrapaolacastrobeltran5655]

@@slcmathpc thanks! :)

[@manojanrangithathas8160]

the last one should be 1 instead of 2 for the third column

[@helloworld-hv9oy]

well done !!u helped us a lot!

[@StaVroula141]

Great job! Thanks!

[@DimitrisGeorgopoulos-w8b]

lineal algebra

[@angeloseconomopoulos8918]

nice explanation, thank you !

[@faithkimeu5542]

good explanation...thank you ...

[@AK-gt8zy]

That was great, thanks so much!

[@esissthlm]

Thank you! Nice video.

[@johnsoto7112]

at 44:25, isn't X5 t = 1? not 2

[@slcmathpc]

The last entry of the third vector should indeed be 1.

[@arunbm123]

nice explanation

[@AMARKUMAR-iu1ht]

Thanks sir for this tutorial

[@hassanalmahdi8358]

thanks a lot , well explained

[@liangchengxi]

Thx, clear enough

[@panazilian]

thanks great video

[@psoma2744]

its worth listening .. thank you..

[@XuEdward]

well explained!

[@Kmano2A]

thank you thank you so much ¡¡¡

[@ThefamousMrcroissant]

You're really difficult to listen to in all honesty, but the explanation is good.

[@RakeshKumar-zw2ey]

Thanks very much sir

[@gulnazalizada]

thank youuuuuu

[@abebayehuabera3644]

thank u that is well video

[@bach1292]

thank you

[@WorldOfDeepThought]

Thx

[@kepplerM]

THANK YOU :D

[@dr.seaaral-dabooni383]

thanks

[@thienhabadao]

thank

[@TheNetflixGamer]

Thaaaaaaank you (:

[@وسامحسين-ف5ك]

ما افتهمت

[@fbzubkfkbnktbnuzktbn]

thank you