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This is known as the Hadamard product and is covered in the video. The tl;dr is that, while it certainly has uses, it doesn't represent multiplication of vector spaces in any reasonable way (in particular, it gives different results when there's a change of basis, while other notions of "product", including dot product, are invariant).

There's a deeper explanation here: https://math.stackexchange.com/questions/185888/why-dont-we-...

1 comments

Maybe that's okay? If you look at regular multiplication, it seems vulnerable to choice of "basis" as well. For example:

6 x 6 = 36

But if we choose 2 as basis, rather than 1, then we have

3 x 3 = 9 (aka 3(2) x 3(2) = 9(2))

But 9(2) != 36

So even regular multiplication isn't invariant under chosen "basis"

If you allow regular real numbers to have a "basis" as shown by your example, you can't simply multiply them either. You treat them as one dimensional vectors and you are back to dot products, cross products, etc.

With regular multiplication, the numbers are the "real" thing you are interested in.

In a vector space, the fundemantal object is the vector. Writing it as a sequence of numbers with an assumed basis is a notational convience.

If you want to define a function for vectors, then you need that function to give the same result regardless of the basis you use to represent the vectors. When you here mathaticians talk about proving that a function is "well defined" this is what they are talking about.

Of course, you might be working with some spefic structure where component wise multiplication is meaningfull and well define. That structure might happen to also be a vector space, and the notation used to write it might happen to coincide with vector notation under the "obvious" choice of basis.

Other specific vector spaces have their own quirky notion of multiplication. For instance polynomials can be viewed as a vector space, with an obvious basis of { x^n }, but polynomial multiplication looks very different from component wise multiplication.

You need the function to give a result that has the same properties, for some specified set of properties, regardless of the basis. GP's question is perfectly reasonable if you haven't already accepted that this set of properties is the only set of properties that matters. The thesis could be better stated "if we define multiplication to have properties X, you can't multiply vectors" which is much less divisive.

I think GP's (and my question) is why those properties and none others are important.