Proof of the Cauchy-Schwarz Inequality

[pure0pwnage]

dont you just love linear algebra? rofl

[SomethingSoOriginal]

Very helpful, thanks.

[bach1229]

I've noticed that many Mathematicians are obsessed with writing different parts of a lecture in different colors. I think it's a OCD thing but it's a small side effect of being a genius.

[riggsrevenge]

@bach1229 Its a teaching strategy to have you remember it better sir. its not just for mathmaticians either, it can be used in any subject, I use it for language, for example if I were to teach english and translate to spanish I would use red for english and blue for spanish, there is lots of research behind it.

It works most of the time.

[Reonaru]

Idol !!!

[alihmod82]

Thank you so much for this clear explanation and I used it to solve a homework

[brotherbob1978]

Can I know what program did you use to the writing of the equations? Many thanks.

[harryz132]

if we only prove it respect to dot product... go back to defination ab(x dot y) = ab(x)ab(y)ab( cos(t)), now ab(cos(t) positive,less than 1, inequality proved... equality iff cos(t)=1 i.e. x is scalar mulitple

in the inner product space than... sure this method is fine...(may be simplified by differentiating quadratic to get t when minimum, and sub t back in to get inequality!

[captainbaked06]

amazing! i wish you were my professor!

[blackphoenix1207]

I'm confused, shouldn't the equality iff cy = x have another part to it, where we start off with just x & y and prove that they're scalar multiples...

[blackphoenix1207]

wait, but the CS equality is an if and only if proof, you've only proved one way assuming that x = cy (where x & y are vectors and c is a scalar) and then plugged it in what about the other way around? Maybe I'm not looking at it right..

[alihmod82]

you can prove it by the same way and suppose that x does not equal to scalar multiple by y for any scalar and then you will get contradiction

[LAnonHubbard]

On third viewing, it's making more sense. My problem was that even though I could understand each step, I wasn't getting any intuition from it. At 16:50 he says in future videos he'll give intuition as to WHY it makes sense. I will rest easy again now!

[LAnonHubbard]

Have to say I struggled a bit with this so I'm going to find another source.

[visaeris]

Omg... this makes things sooo clear...

[sponsoredwalk1]

This proof is ridiculously complicated and unintuitive, which is rare for a khanacademy video. I couldn't find the trick website people are talking about but I checked my book, "Introduction to Linear Algebra" by Serge Lang & he gives the easiest & most intuitive proof there probably is. What does he use? You guessed it - the pythagorean theorem. Get this book and read the first chapter, you'll get a whole new look at how to justify everything from fundamentals.

[inder6uc]

@sponsoredwalk1 Yes if you limit yourself to 2 dimensions then pythagorean theorem intuitively explains it. Sal's proof is generic - it doesn't make any such assumptions and it is true in n dimensions

[sponsoredwalk1]

@inder6uc

Check the n-dimensional Pythagorean theorem that is proven

(page 22) in Lang's book before proving Cauchy-Schwarz (page 27).

[thyorison]

There used to be a website called ""Little Trick to Prove Cauchy-Schwarz's Inequality". The website is still there, but some jerk hacked into it and destroyed it. Some people are just evil scum.

[freezingbeast]

Isn't this much much much easier to prove if you introduce the dot product and do an inequality equation with -1<=cos(x)<=1 and x (dot) y?

[regingwapo]

you didn't prove the "only if" part

[frr]

@regingwapo: The only if part are the assumptions. Assumptions are usually given as true and thus doesn't need to be proven.

[frr]

@regingwapo: The only if part are the assumptions. Assumptions are usually given as true and thus doesn't need to be proven. For example, he/she is human only if he/she is a man or women. That only if part is given to be someone either male or female and unnecessary to prove.

[regingwapo]

@frr Only if means that it can't be something else, so you have to prove that that something else doesn't produce the same conclusion.

[thyorison]

There is a shorter, simple, more direct & natural proof. Google for "Little Trick to Prove Cauchy-Schwarz's Inequality". In general, Khan's Linear Algebra course is the best I've been able to find, but it's always good to look for another viewpoint. Whenever you look at something from a different angle, you get a deeper understanding

[ToKokMa]

To prove a quadratic function is greater than 0 with all real t, knowing that a>0, why not use the discriminant and set it <=0? That would get you b^2-4ac<=0 immediately=)

[kadiyala1987]

The quadratic at^2-bt+c has a min. value of 0 which is obtained when t=b/2a. So when you substitute t= b/2a, shouldn't at^2-bt+c = 0, rather than at^2-bt+c >=0 ? Instead, if you assume t to be any other value its assured that at^2-bt+c >=0 but I am not sure if that would help prove the inequality.

[arj007sin]

Thank you so much.Very intuitive.

[eirikhj]

To me it seems as if you're just proving the inequality when (b/2a). Why can't b^2>=4ac if you plug in another value?

[tonmandude]

Man, this stuff is so hard, I don't know how to do my math hw ;_;

[roth66ro]

b/2a is not random. -b/2a is the x-coordonate of the P(x,y) of a quadratic polynomial, where P stands for the vertex of the parabola. So he took -b/2, because if that inequality stands for -b/2, than is obvious that it holds for any other value.

[InfiniteVolume]

Mr. Khan, don't you have to prove that the inequality is equal IF AND ONLY IF vectors x and y are linearly dependent, i. e., I don't think you proved that there is no other way for equality to occur.

Still, amazing video, I learned a lot.

[nTerSF]

@InfiniteVolume lol.

[jpfry]

Awesome proof!

I think an argument with the discriminant would be a little more natural, rather than plugging in b/2a which takes a bit of foresight and might seem arbitrary.

For those asking, the Cauchy-Schwarz inequality is extremely useful in Linear Algebra, Analysis, and probability. A good amount of proofs in mathematics use this inequality (even in physics too, where CS is used to arrive at the Heisenberg uncertainty principle).

[mufc4everch]

@jpfry Yes, It's also much shorter. But regardless this is good

[l2theaura]

Really you should substitute -b/2a but because of the manipulation, b/2a yields the same result. The reason is because the resulting quadratic came from a binomial squared, meaning it only has one x-intercept, which is where the vertex lies and since a is positive the parabola opens upward. Therefore, P(-b/2a) would yield a result greater than or equal to zero.

[yynotx]

not to be critical, but i wish there was more discussion about what the inequality means before the proof--

[mortemdei]

you're pronouncing "Schwarz" wrong ¬¬

[LWRD]

How to know to put in b/2a ? (at 7:55)

[click4nat]

I guess it is just necessary for the proof.

[laag4]

It's a max/min thing, if you're searching for a vertex on a parabola, it will give you the y coordinate. (I think, it has been a while since I've really done that.

[TheEarlOfDublin]

i believe it was just an arbitrarily chosen value

[nadzTube]

Sal -- are you dedicated full time to your academy?

[Budisawsome]

he is now

[Waranle]

Thank you Sal!

[jakubnage]

Hello Salman Khan. I love your videos and have used them for calculus. You are truly an asset and valuable resource to any student.

The main purpose of this comment (if you see it) is to ask what you recommend I use with my wacom bamboo tablet whilst taking notes in class. In terms of functionality, my only prefernce is that I could have typewritten text together with my tablet writing which would be diagrams.

Anyone who feels like answering this, comment here or message me

[ose90]

Awesome!!! Waiting for the next part which would involve the angle between the vectors

[DrJobinsp]

does any1 know what software he uses to make these vids?

[shifterdude647]

awesome