Inner Product Spaces

[@drpeyam]

Typo: The dot product on polynomials I showed is not a dot product. What I meant to say is this: let your space be Pn (the polynomials of degree n or less) and choose n+1 distinct points xi, and let p . q = sum over i of p(xi) q(xi). That’s a dot product

[@amicm4935]

You know a book that shows the of this one?

[@eswyatt]

@@amicm4935 David Lay's Linear Algebra book uses this example in the chapter on Inner Product Spaces

[@coefficient1359]

I really needed this, thanks for uploading

[@loganreina2290]

Dr. Peyam, shouldn't we have u•v=(v•u)* for full generality? I fully agree with your definition for Real Inner Product Spaces though. Thank you.

[@eswyatt]

Thank you Dr. Peyam. I thought this multiplying two polynomials evaluated at t1, t2, . . . tn, was just a David Lay fetish. Couldn't understand why we weren't multiplying coefficients of the two polynomials. But seeing it as sampling makes it relatable, somehow.

[@eliavrad2845]

And thus did quantum mechanics sprang into existence

[@Theraot]

If we define the angle between two functions using this, and use it compute the angle between two functions that correspond to lines, does it gives you the angle between the lines (perhaps with some tweaking that can be generalized)? Because, that would be cool.

[@drpeyam]

Yep

[@holyshit922]

Why weight function is missing in 7:11 inner product example?

[@drpeyam]

? There is no weight function here in this example

[@gamedepths4792]

Can we use these properties on non uniform circular motion ?

[@loganreina2290]

Also, Peyam, what do you mean by "most general dot product?"

[@alinajmaldin]

When taking dot product of functions, what happened to the “dT” when you go to the summation notation? Thanks

[@muratdastemir1613]

Did you find an answer?

[@aneeshsrinivas9088]

Does the inner product for functions have any visual interpretations. Also is there a cross product for functions.

[@timka3244]

Why you are not uploaded a video on about 3-4 days, dr Peyam?

[@JamalAhmadMalik]

We just started this at university. Please, do more videos on Vector Analysis.

[@drpeyam]

There’s a whole playlist on orthogonality and one on vector calculus

[@JamalAhmadMalik]

@@drpeyam Thanks a lot for the reply, Dr. But we are studying God knows what Talking about grads and divergence, etc. And the properties thereof. It is way over our head--at least by the way we are taught! :/

[@drpeyam]

The vector calculus playlist takes care of all that :)

[@DargiShameer]

@@drpeyam make videos on Hermite and Lauger differential equations

[@Anony1176]

Can you say why summation of a(ti)b(ti) is the integration of ab dx? I couldn't understand the inner product formula at hilbert space. If you kindly tell it will be really helpful for me. Thanks.

[@drpeyam]

It’s not a definition, it’s an intuition.

[@Anony1176]

I think it is because we are assuming that f(x) will remain f(x) on the interval dx.So basically we are adding y's. And as in the interval dx, y will remain y or f(x) so when we are adding f(x) within the interval dx. Total additive values of f(x)'s is f(x)dx. And so whole additive values will be integral f(x)dx. I am a physics research student it is a very important concept in quantum mechanics as bra ket notation. So if you kindly tell me or which theory or book may I read to clear this concept that will be really really helpful for me.

[@academicalisthenics]

Does the result of a dot product always have to result in a single number? If so, wouldn't this also count as a kind of property then?

[@drpeyam]

It’s part of the definition, as a function from V x V to R or C

[@Tomaplen]

when you are not 100% sure about your anser in an exam: Answer "=" 6

[@Jnglfvr]

Your formula at 11:21 is not correct. The angle between t and 1 is the ratio of the integral of t*dt divided by the product of (the square root integral t^2*dt times the square root of integral of dt)

[@pco246]

In the example from the polynomial space you could get a vector which is orthogonal to itself but isn't 0 (for example, x). Doesn't that mean that it's not a true inner product?

[@drpeyam]

See pinned comment

[@Handelsbilanzdefizit]

How do I find a function, that, when it's integrated, the upper Interval-limit is the root of its own integral? Integral_[0,sqrt(F(x))] f(x) dx = F(x)

[@drpeyam]

Oh, I think when you differentiate that and use the Chen Lu, you get a differential equation, which you can (in theory) solve

[@Handelsbilanzdefizit]

@@drpeyam Ok, then lets do a functional equation: f(x)^f(x) = sqrt(f(x)) what's f(x) ? Functional equations are more interesting than differential equations

[@drpeyam]

I’d take ln of both sides, cancel out by f, and then solve for f

[@Handelsbilanzdefizit]

@@drpeyam I know. This was just an example. Solution is f(x) = 1/2 & 1 of course f(x)^f(x) = f(f(x)) is not so easy any more I tried to say, lets discuss "functional equations" Usually, functional equations are more difficult than Differential Equations, in my opinion.

[@Handelsbilanzdefizit]

f'(f(x)) = f(x)² --> f(x) = x³/3

[@Royvan7]

i am curious if there is a good way to phrase the integral definition of f . g is terms of a matrix. i suppose you could have f and g be infinite dimensional vectors and the matrix be a I*dt but that doesn't seem very useful. i suppose if you constructed a basis of function then you could probably construct a useful matrix to represent the dot product. hmm...

[@drpeyam]

Not really, since functions are infinite dimensional (and in fact uncountable-dimensional)

[@xuanchen3434]

I think the analogy of matrix should be a function with 2 variables. Then the generalized inner product can be something like int_x(int_y(f(x)M(x,y)g(y))). However the analogy of "transpose" is still unclear...

[@dhunt6618]

No pun from me today, since this is really cool!!! As an aside, this would be interesting if it could be added to the C++ boost or STL libraries, if possible (assuming it isn't already there). Thanks!

[@dueffff]

Doesn't the dot product for polynomials given here violate the property "U*U=0 => U=O"? since taking a polynomial p without constant term yields p*p = p(0)*p(0) = 0, but p need not be the zero polynomial.

[@drpeyam]

You’re right, should evaluate it at 3 points or more

[@dueffff]

@@drpeyam But the argument still holds as long as we use a fixed finite set of sampling points. We could pick a polynomial with zeros at the given points and its dot product with itself is 0.

[@drpeyam]

I meant to say that if your space is Pn, then it’s enough to have it at n+1 points

[@Rob-J-BJJ]

thank you doctor

[@gamedepths4792]

We need more videos on vector calculus please!

[@drpeyam]

There’s a whole playlist on vector calculus

[@MrNotinthemood]

Lovely video! Could you perhaps do some videos on Topology, maybe some tricky compact space proof? Greetings from Sweden!

[@drpeyam]

Check out my analysis playlist

[@justins.2138]

Thanks!

[@Handelsbilanzdefizit]

Dot product of functions reminds me of quantummechanics. Sometimes, there's a discrete spectrum of eigenstates, like the two-dimensional "spin-phase-space" (↑ or ↓) Sometimes, there's a continuous spectrum with infinite possibilities, like the position of an electron in 3 dimension. Or 4 dimension??? Who knows? In short terms: Electron spin has "two" endstates & electron position has "infinite" endstates. But It's just lame physics, not maths ^^

[@willnewman9783]

I believe the one you defined for polynomials does not work. For any polynomial with p(0)=0, we have p dot p =0, but p itself doesnt have to be 0

[@drpeyam]

Yeah, but something similar works for Pn if you evaluate it at n+1 points

[@zuccx99]

What up early.

[@Tomaplen]

a.k.a. Metric Tensor