Least squares approximation | Linear Algebra | Khan Academy

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The least squares approximation for otherwise unsolvable equations
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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.
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[@sarabenavidez3863]

"Some of you might already know where this is going.." Me: Nope

[@howbani]

Hahaha

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What do u mean

[@aashaytambi3268]

When I get a real job, I will donate my bonus to Khan Academy. This has saved me so much time and you are so awesome.

[@Kialikhun123]

Did you get a job yet?

[@mafiafiazmafiafiaz3410]

Bol na jana❤❤

[@user-ew2sz3ez4n]

They are still waiting fir your bonus mate

[@_spectre_1589]

This lesson is fantastic! I understood the problem in only 15 minutes! You're absolutely better than my numerical analysis teacher at university, that can't properly teach an argument in two hours! Thank you!

[@haojiang4882]

Comes in handy while studying machine learning.

[@KeystoneScience]

yes, same

[@mwaleed2082]

Very true. When I was studying ML, "normal equation", I really thought that I had seen it somewhere. Then I realized I studied it in Lin. algb.

[@adamhuang2421]

very helpful! Thanks a lot! you are doing great things! I also listened to your other videos, all very wonderful!

[@nicholascunningham6936]

This channel is a blessing. I've had some really bad professors and I've had some really good professors. But even the really good professor never made the concepts click with me as well as these videos do. Like, not only do I understand the math better, but just the little diagram you drew showing A's column space, and visibly showing how b is outside of A's column space yet could still be approximated using a vector v in A's column space, like idk how else to describe it but that just made it click for me. Edit: I guess _one_ way to describe it and how it clicked for me: So we use a line to approximate a bunch of data points on a graph, or plane. If these data points were in a straight line, the "approximation" would have no error. However, this is often not the case. Now, think about the equation y=mx+b. Let's use c instead of b to avoid confusion in the next step. So we have y=mx+c. This is the equation used to represent our line. Suppose b=y-c. Then we have mx=b, which looks a lot like Ax=b. And it is! A is just a 1x1 matrix. So the line is bounded by the column space of A, or m, and our variable(s) (in this case, just x) can be changed to get b. Just basic algebra: if m=3 and b=6, then x=2. But say b is a 2D vector, e.g. b=(1, 2)^T. Well now, no matter what x you use, you can't get b (unless b just happens to lie on the line). You can only get as close to b as the column space of A will allow you. In the diagram drawn in the video, the column space of A is a plane, so the span of A is 2. For simplicity, let's suppose A is a 3x2 matrix (a geometrical interpretation of this is that A is a 2D plane "floating" in a 3D space). b appears to be a 3D vector (so while A is only a 2D slice of the 3D space, b is a point that could be anywhere in the 3D space). So, just like before, we try to use a line bounded by the column space of A to get as close to b as possible by changing our variables (in this case, x1 and x2). Correct me if my understanding is wrong :)

[@rajj1567]

Your videos are just great !!! The concepts with geometrical examples make very good sense !!! Thanks a lot

[@fireheart9715]

This was incredible, I started this video off being so confused about the least squares, and I just get it entirely now! Thank you so much :)

[@samirrimas]

Very useful man you are doing an amazing job this literally saved me hours of searching and reading can't thank you enough :)

[@elmiramb]

Thanks a lot, very comprehensive ! great job!

[@xesan555]

Thanks so much Khan...wonderful explanation in two videos that explains everything...great. You are wonderful

[@ozzyfromspace]

I was doing an online machine learning course and got lost when the lecturer introduced the normal equation (which this is, with a different name). Needless to say, I'm finna binge-watch your linear algebra lectures now because I get insecure about using equations I don't understand. Thanks for the playlist, I really wanna put ML in my toolset so we're doing this!

[@khaledsherif7056]

Can you please mention the name/link of the course ?

[@mwaleed2082]

@@khaledsherif7056 not sure which course he used for ML, but I'm studying Machine Learning by Andrew NG on Coursera. When he was teaching us normal equation as an alternative to gradient descent In Week 2 of the course, I realized I had seen this in Linear algebra but with a different name which is the title of this video.

[@ottoomen5076]

Excellent explanation of a valuable technique.

[@jibran6635]

This is super useful in solving assignments.THanks khan academy.

[@aidawall8]

You are like a billion times better than my professor... and my professor isn't even bad. On the contrary he's my favorite! You're just even better at explaining things. Plus it's impossible for me to lose focus with the pretty colors and your beautiful handwriting. lol I have my Linear Algebra final tomorrow (technically today) and I owe the A that I'm sure to get to you and all your helpful videos!

[@potatojam6519]

7 years later... did you get an A? :)

[@HustleHeaven247]

11 years later did you get that A?

[@johnfykhikc]

Best approach to the problem. No gradient, no multivariable calculus. you're master!

[@Jshizzle2]

Helpful exploration of least square properties

[@user-ot4rp8yn8r]

Best linear algebra playlist.

[@kartarsingh7776]

Super clarity......

[@Awhobiwom]

Thank you so much. You just simplified long boring hours of confusing lecture

[@seprage]

It would be great having links when says "I explained (whatever) in a different video" to access that explanation. In this case I wanted to know why C(A)transpose=N(Atranspose). Thanks¡

[@Daski69]

+Sergio Prada same thing here

[@196phani]

www.khanacademy.org/math/linear-algebra/alternate-bases/othogonal-complements/v/linear-algebra-orthogonal-complements go through this to understand how C(A)transpose=N(Atranspose).

[@MrVishyG]

+1

[@gulshanjangid3470]

consider any vector x perpendicular to Column space of A i.e. belongs to A _|_. Then dot product of A and x is 0, i.e. (A^T)(x) = 0 Now consider b = A^T, so clearly above equation is bx = 0, i.e. x lies in null space of b Thus x lies in null space of A^T also as in the first line I said x belongs to A perpendicular , thus C(A _|_) = null(A^T)

[@fascist27]

really helpful

[@rbfreitas]

Good video!!!! And nice work! Good luck with the KhanAcademy :)

[@tranzconceptual]

god dang it I knew I should have chosen other bachelor thesis..

[@user-xedwsg]

haha!!!!

[@OurEverlastingYouth]

just realizing this now as well

[@rob6129]

first semester stuff at my uni

[@gangigooga7710]

@@rob6129 what uni u attending?

[@MistrVahag]

Excelent video. Thanks much :)))))))) Vahag

[@mattralston4969]

Thank you Salman Khan. I appreciate the opportunity to relearn the method here. You can never hear this stuff enough times.

[@batmendbatbaatar4290]

This is surprisingly easy

[@rob6129]

Nice derivation of the normal equation

[@kalvinsackey1804]

can we please get a video for the maximum likelihood estimation

[@Matterhorn1125]

can you teach me cubic expressions and cubic equations :) eg. solve the equation x(3X3X3) - 2x(2X2) - x + 2 = 0 by using the factor theorem formula :)

[@adithyavarma758]

thank you very much sir

[@lancelofjohn6995]

It seems I have seen the best video!

[@inserthere6387]

great geometric intuition of linear regression

[@EWang-yn5sy]

This guy is good...........

[@mustafasabeeh8893]

thanks

[@arico94]

Should have used n instead of k its usually mxn in R^n

[@ArafatAmin]

what happens when AT*A is singular. How do we solve for the least square solution?

[@budharpey]

Very useful! In my lecture slides I had this term Hx=z for the same problem and I couldn't make sense of how we could get to this as the best solution: x = (Ht*H)^-1 * Ht * z. Now I understand:-)

[@luffy08dn]

thaks

[@zhiqiguo803]

love this guy

[@BlackfireGippal]

I wish to know how to solve this: x has values of : -2 0 1 2 3 and y : 17 5 2 1 2 and i'm asked to use the least squares method, but i've been absent and i don't know exactly what my teacher ment by that or what that method consists of. Can anyone help me solve this ?

[@SanwaOfficial]

I have one question, whether the LSS always consistent? if yes, how can I prove it? please answer

[@mwaleed2082]

Hi, not sure if you're still looking for the answer, but could you please describe what do you mean by consistent?

[@bb_-.-_-.407]

It means that wheather we can always find least square solution of a system.

[@utte12]

nice vid, but why did you take the length squared? i understand that the length of the vector would be sqrt(b1^2 + b2^2...bn^2) but why did you square even that?

[@lucasm4299]

utte12 Because it’s easier to work with minimizing the sum of squares than minimizing the square root of a sum of squares. That’s my guess

[@trejkaz]

I tried using this trick for the problem I'm facing, but it turns out that when I multiply AT by A, I get a matrix which isn't invertible, so I still can't solve it. LOL This _still_ seems odd to me, because even if some element in the input matrix A was contributing 0 to the result b, it should _still_ be possible to get a point as close as possible to the result.

[@MrZulfiqar37]

I have a question.. does least sequare approximation has always solution..

[@CR-iz1od]

+Zulfiqar Ali not if you don't solve it.

[@Daski69]

+Conor Raypholtz it still has a universally reasonable solution

[@spindash64]

I'm pretty sure that is the idea of least squares: to provide a close answer when you can't give an exact one

[@shredding121]

it does always have one - if Ax = b has a solution than it's a vector on A and if not it's the projection on A.

[@natebush26]

There is always a solution to the least squares problem. Why? x* is in colspace(A) by definition of being a projection from b into C(A) so there must be a set of weights that yield a linear combination of a that equal b.

[@winnies1001]

how did you know that it was a projection to the Col(A) and not anything else like the Range(A)?

[@lucasm4299]

Winnie Shi Col(A) already is the range of A.

[@kavishdoshi2408]

accha hai

[@user-te2hd9nb8q]

당신은 나의 구원자입니다. 정말 명쾌한 강의입니다. 감사합니다!! 👍👍👍

[@shinigummyl1586]

2018? Im alone :(

[@diesel7777777]

I'm here.

[@bli240]

Onto 2019!

[@bb_-.-_-.407]

2024 here

[@priestofrhythm]

I am the 60th guy liking it !! :P :D Great vid, thank you. :)

[@91leonetammie]

This is the first Khan Academy video I watch and don't understand...

[@mwaleed2082]

For that you need to study orthogonal components, and the concept of what spanning sets are which further derive the concept of column space, null space, etc.

[@nilsclaessens5203]

❤

[@Ben.N]

Big brajn

[@spechtbert]

n1

[@dion9795]

bro just do an example lol

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🤩

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Sometimes I can't see what he's writing.

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gorgeous

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[@fascist27]

Respond to this video...

[@roy5180]

thank you sir