Proof of the Cauchy-Schwarz Inequality

Proof of the Cauchy-Schwarz Inequality

Category:

Education

Tags:

• cauchy
• swarz
• vector
• dot
• product
• length
• linear
• algebra

[regingwapo]

you didn't prove the "only if" part

[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).

[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.

[pure0pwnage]

dont you just love linear algebra? rofl

[SomethingSoOriginal]

Very helpful, thanks.

[mufc4everch]

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

[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.

[Reonaru]

Idol !!!

[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

[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!