Proof of the Cauchy-Schwarz Inequality

sponsoredwalk1 Says:

Oct 11, 2010 - 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.

visaeris Says:

Oct 17, 2010 - Omg... this makes things sooo clear...

LAnonHubbard Says:

Dec 8, 2010 - Have to say I struggled a bit with this so I'm going to find another source.

LAnonHubbard Says:

Dec 8, 2010 - 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!

frr Says:

Jan 21, 2011 - @regingwapo: The only if part are the assumptions. Assumptions are usually given as true and thus doesn't need to be proven.

frr Says:

Jan 21, 2011 - @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 Says:

Jan 23, 2011 - 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.

blackphoenix1207 Says:

Feb 6, 2011 - 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..

blackphoenix1207 Says:

Feb 6, 2011 - 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...

inder6uc Says:

Mar 6, 2011 - 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 Says:

Mar 6, 2011 - Check the n-dimensional Pythagorean theorem that is proven(page 22) in Lang's book before proving Cauchy-Schwarz (page 27).

captainbaked06 Says:

Mar 15, 2011 - amazing! i wish you were my professor!

harryz132 Says:

Jun 15, 2011 - 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 mulitplein 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!

brotherbob1978 Says:

Aug 8, 2011 - Can I know what program did you use to the writing of the equations? Many thanks.

alihmod82 Says:

Sep 30, 2011 - Thank you so much for this clear explanation and I used it to solve a homework

alihmod82 Says:

Sep 30, 2011 - 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

Reonaru Says:

Nov 21, 2011 - Idol !!!

bach1229 Says:

Dec 1, 2011 - 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.

mufc4everch Says:

Dec 22, 2011 - Yes, It's also much shorter. But regardless this is good

SomethingSoOriginal Says:

Jan 10, 2012 - Very helpful, thanks.

pure0pwnage Says:

Jan 27, 2012 - dont you just love linear algebra? rofl

riggsrevenge Says:

Feb 15, 2012 - 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.

SCArmCannon Says:

Mar 10, 2012 - I think that it's more convenient to just use the identity that the dot product of any two vectors is their magnitudes multiplied together, times cosine of the smallest angle between them, to prove the inequality, or the equality when they are parallel vectors since theta (smallest angle) is 0, so cos0 is 1. For any other non-parallel vector pair, since costheta has the domain between neg 1 and pos 1, then we divide both sides by the magitudes to get 1 >/= costheta, which is true.

regingwapo Says:

May 6, 2012 - "The only if part are the assumptions" - nope, they're not.