Inner Product Spaces

Inner product space and examples. It's like a generalization of dot products to functions! Examples of projections and Gram-Schmidt to 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

[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.

[coefficient]

I really needed this, thanks for uploading

[loganreina]

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.

[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)

[eliavrad]

And thus did quantum mechanics sprang into existence

[holyshit]

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

[JamalAhmadMalik]

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

[dhunt]

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!

[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.

[gamedepths]

Can we use these properties on non uniform circular motion ?

[alinajmaldin]

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

Thanks

[loganreina]

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

[aneeshsrinivas]

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

[timka]

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

[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?

[MrNotinthemood]

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

[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.

[gamedepths]

We need more videos on vector calculus please! <3

[pco]

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?