>>12380568true. but in that case the vector axioms are redundant and things can be described by the underlying field alone.
>>12380600>They are not represented by vectors in R^2, they are in the same equivalence class.it is representation, and representations can have additional structure. you can form equivalence classes between representations, though
https://en.wikipedia.org/wiki/Representation_theoryi've been studying multivariate complex calculus and wirtinger derivatives recently for optimization. they are interesting because they blur the line between R^2 and C. for optimization you need to take complex partial derivatives, but complex-differentiability is an extremely strong constraint on a function because it requires the function to satisfy the cauchy-reimann equations. moreover, most cost functions are real-valued complex functions (norms are popular), which are not complex differentiable because they don't satisfy the CR equations (unless the function is a real constant, which isn't useful). so how do you solve this shit? you take advantage of the additional structure of R^2 and take the partials of z = u(x,y) + i v(x,y) w.r.t the real variables x and y because these don't have to satisfy the CR equations. this works because partials=0 are a sufficient condition for a critical point. from this you can develop a complex gradient and apply gradient descent or krylov methods (i.e. conjugate gradient). analyzing the wirtinger derivatives a little more in the context of complex-differentiable functions, it is seen that a complex number can properly be viewed as a single number
