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JamesGold
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Every resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?
Consider the set ##\mathbb R^2=\{(x,y)|x,y\in\mathbb R\}## of ordered pairs of real numbers. If we define the addition of two arbitrary members of this set byJamesGold said:Every resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?
JamesGold said:To what extent are they arbitrary?
The best one can hope to prove is that there is a vector space in the branch of mathematics defined by this other set of axioms. If you just supply the missing details from what I said about ##\mathbb R^2## above, you're almost done with such a proof. You could take the definition of the real numbers and a few set theory axioms as your starting point (your axioms need to e.g. guarantee the existence of functions and cartesian products), or you can take the axioms of ZFC set theory as your starting point and start by proving the existence of a Dedekind-complete ordered field (i.e. a set whose members have all the properties listed in the definition of the real numbers).Stephen Tashi said:If you want to find a lower level mathematical system that can be used to prove the axioms of vector space, I don't know of any work that has done that, but someone else on the forum probably does.
A vector space is a mathematical structure that consists of a set of objects (vectors) and operations that allow for the addition and multiplication of these vectors. It is used to model and analyze various mathematical concepts, such as geometry, physics, and linear algebra.
The vector space axioms are a set of properties that define the fundamental characteristics of a vector space. These include the closure under addition and scalar multiplication, associativity, commutativity, and the existence of a zero vector and additive inverse.
The vector space axioms were first introduced by mathematician Giuseppe Peano in 1888. They were later refined and formalized by mathematicians such as David Hilbert and Emmy Noether, and have since become the standard set of properties used to define vector spaces.
The vector space axioms are important because they provide a framework for understanding and analyzing vector spaces. They allow for the development of consistent and rigorous mathematical theories and provide a basis for solving practical problems in many fields, including physics, engineering, and computer science.
Yes, the vector space axioms can be modified or extended to include additional properties or to apply to different types of vector spaces. For example, in functional analysis, the vector space axioms are extended to include a norm function, which measures the length of a vector. Additionally, different types of vector spaces, such as complex vector spaces or infinite-dimensional vector spaces, may have variations of the standard axioms.