[gpm]

Defining "multiplication" to be "any function that takes two arguments and outputs one" isn't a very interesting definition of multiplication. The word is a lot more useful if you put constraints on it like saying for something to be a multiplicative operator it has to respect (a + b) * c = a * c + b * c (the distributive property).

Once you put a "reasonable" set of constraints on it... you discover that you can't actually multiply vectors (no function exists that satisfies the properties you want). Though the talk isn't about proving that (or justifying the set of constraints that mean you can't multiply vectors) and instead goes off in another direction of extending your vector space to a bigger space (like how the complex numbers are a bigger space than the reals) where you can define a reasonable multiplication operator.

[dventimi]

Doesn't the scalar product have the distributive property?

[gpm]

Yes, that was an example of one property you probably want, not a set sufficient to make it such that no such operator exists.

Another property you want (and the talk uses) is that the operator is that the operator is from V x V to something. I.e. we are multiplying two vectors (because that's what we asked for in the title) not a scalar and a vector. That excludes your counter example, but still isn't nearly enough to make it so that no multiplication operator exists.

I'll be honest and say I'm not listing properties here because I don't remember what properties are needed to make it so you can't define the operator... hopefully someone who has studied this a bit more recently or thoroughly than me can chime in.

[empath-nirvana]

Basically you want all the properties of a ring:

https://mathworld.wolfram.com/Ring.html

Scalar and dot products don't stay within the group of vectors and component wise multiplication doesn't always have inverses.

[gpm]

You need slightly more than being a ring. It's possible to make a ring (even a field) over R^n provided you don't care about interactions with scalars. For example: Just take any one-to-one map from R^n to R and then apply the operations in R before mapping the results back. It won't make any geometric sense, but it will be a ring.

I've been blanking on what exactly the interactions with scalars that we need to preserve are...

[bashmelek]

If I remember correctly multiplication requires a “zero,” an “identity,” and for something to be a field each item needs an inverse. I imagine we can define multiplication in R2 just as we do for C. So by that logic we ought to be able to define such an operation on any R(power of 2).