Monday, May 22, 2023

The Limitations of Grant Sanderson's Math youtube channel: 3Blue1Brown? A deep dive into complex analysis?

 

Complex multiplication is of course super important, and multiplying by a complex number corresponds with a certain linear transformation (some combination of scaling and rotating to be specific). Cross products, dot products, and complex multiplication are all just different operations between 2d vectors that are useful for their own reasons. The reason complex multiplication is usually treated separately is because it gives much more structure to the plane. It's more analogous to the kind of multiplication by real numbers that we're used to, and it's actually something you can do algebra with (e.g. xy = 0 implies that one of x or y must be 0, which is not true with dot or cross products). As a result, one usually dives deep with the implications of this structure in a class like "complex analysis", and for whatever reason, the connections between this structure and linear algebra are not typically talked about within a linear algebra class itself.
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3Blue1Brown Thank you for your answer.
Or you could get your mind blown by quaternions (or the algebra of physical space in my preference) and how they not only unify dot- and cross products, but complex multiplication as well! These algebras have indeed the property of conserving the 'length' of multiplying two vectors. The interesting part is that quaternion algebra was the way to do spatial calculations, before being displaced by vector analysis, and our notation for the unit vectors i, j and k come from this algebra.
RealCottonCandyKid So quaternions were the original matrices. Cool didn't know that. But I still don't understand the connection between complex algebra and vector analysis. I don't get how you would use quaternions instead of matrices, and if you did how would that relate to cross/dot products.
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A quaternion is made up of a scalar and a vector part. If a quaternion has a zero scalar part, it's a vector, and if it has a zero vector part, it is a scalar. The quaternion is then just the algebraïc sum of these two parts. Quaternion algebra is defined by its (noncommutative) multiplication. If you take this multiplication of two vectors, say u and v, then the dot product of those two vectors is equal to minus the scalar part of that multiplication, and the cross product equal to the vector part, so u · v = - S( u v ) and u ×; v = V( u v ). The quaternion multiplication of these vectors is then just the difference of these products, so u v = - u · v + u × v. This is a very short introduction, and it might seem weird where those minus signs come from, but these disappear when considering paravector algebra (but introducing an imaginary unit multiplied by the cross product, defining it the wedge product).
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 @RealCottonCandyKid  Do you know if Grant Sanderson discusses noncommutativity in his videos at all? Quantum physics professor Basil J. Hiley says this is the secret of nonlocality based on the Jordan Product. There is a new quantum Ph.D. channel inspired by Grant Sanderson and that channel has a superposition vid stating that the noncommutativity is nonlocal - but he doesn't go into the math details as much. thanks, drew
 
 
 

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