The discussion centers on the relationship between vector subtraction and topology, specifically in the context of vector spaces. Participants clarify that vector subtraction is defined as the addition of an additive inverse, which is consistent across all vector spaces. The conversation highlights that while vector addition and subtraction are fundamental operations, the confusion arises from the misunderstanding of these operations in relation to topological properties and their definitions. It is established that subtraction is not treated as a separate operation in algebraic systems due to its non-commutative and non-associative nature.
PREREQUISITESStudents of vector calculus, mathematicians, and anyone interested in the foundational concepts of vector spaces and their operations.
fresh_42 said:Could you be more specific? What do you mean by operation of vectors?
Vectors are usually defined as an additive group over a field.
This means we have addition, subtraction, associativity, zero element, and distributive multiplication by e.g. reals, or rationals. There is no topology at prior, although there are topological vector spaces. They play an important role in physics. But whatever vector space we consider, vector addition and subtraction is fundamental for all of them. It is why we consider it in the first place.
This is true. The additive inverse element of a vector $$v$$ is $$-v$$ and $$w-v=w+(-v)$$ is the vector addition. We cannot divide vectors!doggydan42 said:I could be a misunderstanding on my part.
From what I understood, we call vector subtraction the addition of the inverse, but it is not defined the same way as it is for real numbers. But this just confused me, as I could not see why that would be the case.
What class is this , for context? Maybe you mean vectors based at different points?doggydan42 said:Summary: Why is the operation of subtracting vectors not defined?
I learned in a vector calculus class that the operation of vectors is not defined. The professor mentioned it had to do with topology. How does the operation of vector subtraction relate to topology and how does topological properties prevent vector subtraction from being defined?
I'm taking vector calculus, but the professor just mentioned that and went on. So I was curious as to why that statement was true.WWGD said:What class is this , for context? Maybe you mean vectors based at different points?
I assume he spoke about another operation. Of course we can only guess, so a few possibilities are:WWGD said:Strange thing is that if v is a vector in the sense of a vector space, then the set of vectors is an Abelian group.
The professor basically said not to bother asking.mathwonk said:what do you suppose would happen if you raised your hand and asked?
Yes, but the OP is about subtraction of vectors not being defined.fresh_42 said:I assume he spoke about another operation. Of course we can only guess, so a few possibilities are:
- no multiplication instead of addition: ##\vec{u}\cdot \vec{v}##
- no addition in GL(V): ##A+B##
- no addition in a group representation: ##(g+h).\vec{v}##
- no addition by scalars: ##c+\vec{v}##
All not allowed (at prior).
I meant that the teacher might have meant something else than the OP means he meant. Don't you mean?WWGD said:Yes, but the OP is about subtraction of vectors not being defined.
They may have been referring to vectors based at different points. Not too clear from the OP.pasmith said:How do you do differential calculus without being able to assign a meaning to f(t + h) - f(t)?
As others have said, that statement is not true. So you need to ask him what he meant or we are left to guess.doggydan42 said:I'm taking vector calculus, but the professor just mentioned that and went on. So I was curious as to why that statement was true.