The Gram-Schmidt process | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy

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Finding an orthonormal basis for a subspace using the Gram-Schmidt Process
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[@baekalfen]

For me, 20 minutes of this is worth more than a 2 hour lecture. Thank you!

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Khan Academy coming in clutch right before my Linear Algebra final #KhanAcademyIsBetterThanNYU

[@HakaTech]

Asia Suarez?? Interesting name. My name is Europe Patel

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You were meant for each other. You need to have a child together and name it Antartica Koulibaly!!!

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and thier grand kids must be barcelona and madrid

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@@metra-it1985 nah turkey would be better

[@greensasque]

Although my linear algebra instructor made this easy to follow in terms of the steps and calculations, watching this video I know actually understand WHAT we are doing when we perform these steps. Thank you Sal for the great video tutorial.

[@lugia8888]

Read your book too. Sometimes videos aren’t necessary to fill the gaps the instructor doesn’t.

[@JT-hl6zd]

This felt so intuitive... my mind is blown. Thank you for showing us why math is so cool!

[@knjiga4]

i've never understood this quite right at college but now as i take only this one 20 minute lecture at khanacademy i understand everything. thank you!!

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I am so confused.

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join the club

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damn your U :p Tutorials are really nice, and that's really helping. Thank you :)

[@ellen128]

This is so well explained. I had to go back watch it twice though. But makes complete sense now. Thank you so much!!!!

[@steveosazuwa2710]

Saving mathematic lives out here, thanks fam

[@Space_Lion]

Man this guy's the best. Been using these vids since middle school, and I'm in grad school now. Thanks Khan Academy

[@theoldblood3804]

Hes definitely right when he said its not that bad when youre dealing with the numbers. Memorize the equations and its not too bad. Understanding the proof and why is the hard part.

[@Iphone-bk2mk]

You're truly great at teaching! Thanks and good luck!

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A day before my optional resit of quantum mechanics and here Khan Academy is, saving the day again.

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+Árón de Siún Here for my test of Mathematical Methods of Physics :p

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thank you khan academy ,it was really useful .i was struggling to understand hilbert space now you made it easy.

[@derek8482]

Thank you so much, i am so grateful to you Sal. You literally changed everything....

[@ruttananrongsawad2232]

It became clear later on in the video. thank you so much for making it free!

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Thanks kahn. You're really helpin me out with my math classes

[@anweshadutta8782]

I don't have words to show how grateful I feel now

[@rishubits]

great sir!! i love listening to your way of explaining things

[@lilymayburke8413]

Explained so well! Thank you

[@noneofurbusns3139]

That awkward moment when you understand something in maths :') Thanksss!!

[@lynny7868]

1:28 How to ensure a basis is orthonormal. 5:55 Replace v2 with the orthogonal projection of v2 onto v1 and the vector component of u orthogonal to v1.

[@putin_navsegda6487]

God bless you Khan! Thank you for your work! 😇

[@vko7059]

Perfect lecture. Thank you.

[@jhonatanhernandez3568]

This explanation blew my mind

[@rkishei]

go Purdue! also taking linear second midterm lol ... Love your videos Sal, very helpful in understanding these abstract concepts that always get muddled in my head by the book.

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Best Explanation EVER!

[@SuperChad1313]

Minus that business right there.......love it.....

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You are my life saver!!

[@rhsee]

You are a life saver!

[@sor715]

hey what program are you using for recording/writing up that stuff?

[@bangthatdrumb]

these vids are great in conjunction with PAULS online math NOTES

[@HoloUniverse]

Much appreciated!

[@serenliceackeric9732]

Thank u so much ! This video really helps me a lots ! by the student from Tw

[@Xcrypt1991]

Wish I could do the same. I don't go to many lectures but I still have to read my books because Khan Academy's math is not the kind of formal proof-oriented math you learn in pure mathematics

[@user-wl9zo2mi4u]

无可想象的伟大作品，足以名垂青史的杰作！！！！！

[@tomasdejmek2520]

You made small mistake, when you copied ( v3 u1 ) u1 + ( v3 u2 ) u2; you created "y3 = v3 - ( v3 u1 ) u1 + ( v3 u2 ) u2" right is, "y3 = v3 - ( v3 u1 ) u1 - ( v3 u2 ) u2", i think.

[@thebugbear]

Yes, the sign changes, or you need to use parenthesis.

[@elu7193]

thanks! i understood

[@kenikozo]

Is there a video from khan that is about "inner product spaces"? Help would be greatly appreciated. Thanks!

[@achillesarmstrong9639]

nice job ,easy to understand.

[@marcmarc1637]

Thank you!!!

[@youssefbenhachem993]

Oh thank you so much !

[@baruahsarthak_]

Superb explanation!

[@sana0hk]

you r my new hero lol! thank you so much

[@amairanas8631]

omg 10 years ago!! how is life goin now?

[@yeubememe2802]

Thank you!

[@pz0utable]

I can't believe it, I'm starting to love Linear Algebra

[@MrBiudsbiribiubu]

Where can i find the bibliography you used for this lecture? Now I've got the idea but i'd like to read it ;/

[@TheNoHAnthony]

Thank you. Things in my linalg book become too complicated when they just throw theorems and lemmas at you without explaining what the formulas actually mean....

[@Vrig]

Hm.. could someone please inform me if this is a correct conclusion: When he takes the dot product between (5min-10min) v_2 and u_1 and then multiplies by u_1 it's because the dot product yields the magnitude of the projection and by multiplying it with u_1 he gets the vector x? If this is the case; wouldn't it be the same thing to take dot product between v_2 and v_1 and then divide by |v_1| ?

[@TheJohnnyPatriot]

Vrig I thought the same thing when I was doing this in my Linear Algebra class 6 years ago. not sure why you're taking the dot product of the unit vector...the thing is, you still have to multiply by u_1 to get the projection because the dot product is just a number, not a vector

[@맥스웰방정식]

thx u so much😊😊😊😊😊

[@linleybaruch738]

Please give my exam for me :( Thanks for the tutorial, great explanation :)

[@TheExoticDarkness]

I don't understand quite why he claims that the projection of v2 on to the subspace v1 can be described as (v2.u1)u1. Can someone help me understand, or is there perhaps part of another video that I missed?

[@AL-jg8pv]

taking the dot product of 2 vectors gives you the product of how much the 2 vectors travel in 1 direction....but since u1 is a unit vector...so you can say the dot product 'v2.u1' gives you the component of v1 travelling in the direction of u1 ....though the dot product is just a number.....to actually specify that it is in the of direction 'u1' you multiply the dot product (v2.u1) by the vector u1 to give it direction..

[@blondii0072]

Thanks man

[@marquez2390]

How do you know everything?

[@nrrgrdn]

minor correction @2:15 he should've written " || u_1 || " instead of just " u_1 "

[@ASTROTZUR]

There is a small mistake at 16:09 the plus in the pasted part should become a minus.

[@gptty]

Doing god's work!

[@xmrwzw]

thaaaanks a lot

[@standardcoder1184]

Do the orthnormal vector basis that we get by Gram-Schmidt are unique

[@lugia8888]

No. Orthonormal bases are not unique.

[@user-xs9oo9gc7u]

I'm quite curious, are the subtitles automatically generated? If not, what's going on at 1:46? lul

[@somebody4061]

He said, "you can argue the zero vector is in there."

[@jennabockman727]

thank you.

[@amerkhoury8034]

I really wonder why not all profs and books explain that the same way

[@nahvkolaj]

I...understand!

[@spechtbert]

whosss the mann??? khansss the mannn

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Khan Academy saving students' asses all over again why can't Professors teach like that in universities?

[@not1AM]

thanks

[@lolalukie713]

well a in this case is normal (length of one) so a . a is just one

[@theoldblood3804]

"Its that easy"...lol

[@user-mu2qq3eb7t]

V_3 is all the linear combinations of v_1, v_2 and v_3, which contains all the vectors in 3 dimension space(assuming it is R^3). in which space does the fourth basis vector v_4 live? Mathematically it's there, but where is it geometrically?

[@anubhavbhura13]

no where. We as humans do not have a spatial 4 th dimension.

[@lugia8888]

After R^3 we do not consider geometry, only structures.

[@zoala001]

i cant get it cuz at school we use a diiferent formula for projection. projF on a :[( F.a) / (a.a)]. a

[@supersonic174]

when you compute the projection, you missed the denominator no?

[@450RacerBill]

Aweri Blakely I’m a little late to answer but the denominator is missing because he’s using the unit vector, I had the same question at first. U = v1 / |v1|

[@valkon_]

very hard...

[@muh_guts]

Sal got color blindness, at 14:25 he meant yellow vector not green

[@danx74]

@redougulas I have second midterm tomorrow

[@StaticSleet]

1.75 speed, thank ye, thank ye

[@padmabatinayak2324]

❤❤❤❤

[@Terrax221]

When i saw the title i was like: "I know Gram-Schmidt, it´s easy", when i finished the Video is was like: "Aaaaaaaaaah, thats how it works!"

[@afmfakhruddin1773]

Proud to be a Bangladeshi!!.........

[@lowzyyy]

Correct me if i am wrong but you are missing something. When u were making projections u forgot that for example V2 onto V1 is equal to (V2 dot U1)*u1 OVER lenght of u1. Why? Because projection of some vector lets say A onto some line s is equal to ||A||cos(angle between A and s) Furthermore if say that A vector is V2 then V2 dot u1= ||u1|| projection V2 onto V1 . So projection V2 onto V1 =V2 dot u1 OVER ||u1|| so u are missing to devide everytime with lenght of base vector

[@ga2yb]

lowzyyy the sizes of u1 U2 etc are all 1 since it is orthaNORMAL. so you can divide by the size, but since it is 2, it doesn't change the outcome

[@lowzyyy]

yes, u1 is 1 and lenght is 1.

[@lugia8888]

u is unit length so you divide by 1. check your calculations and think about it some more

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not all heroes wear capes.

[@crossiantlover]

me too :P

[@yxooo]

@TheNef77 same story

[@Ayplus]

SCHMIDTTY!

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IS there no audio?

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i love you

[@antonblue11]

@sherajr me too lol

[@thesparkflyer]

Gram-Schmidt is the shit

[@karthikeya0804]

he thought for the future 11years ago

[@sherajr]

You are the man! Saving my ass from my shitty professor!

[@internet_user1131]

Why are the concepts of math always made so hard? Why they can't be shown in simple terms like in this video? Who benefits?

[@lugia8888]

Some instructors are simpler than this video too

[@lolalukie713]

I started clapping. :')

[@bangthatdrumb]

1 professor disliked this video as it made him look bad.

[@TheNef77]

@sherajr You must be in my class. I hate that lady.

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You're just spitting English man 😅

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im cooked bruh

[@redougulas]

anyone from purdue? i hate linear!

[@camilly0902]

Haha you repeat things a lot, like individual words :)

[@patmaxable1]

9 years ago :o, how is life treating you now?

[@patmaxable1]

@dragonzito mojado It all went downhill