The Real Definition of a VectorSpace  

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My last post risked being confusing, so here is a more friendly and intuitive article:

 

What are vectors? You might think they are like points, except with a magnitude and angles and such. What are vector spaces? You might think they are subsets of ℝn, where adding two vectors in the space gives you another in the space and all scaled values of a vector are in there. What if I told you that those things aren't vector spaces? At least, not yet.

Let's clear this up.  ℝn is indeed a vector space given the operations, but someone just looking at n-dimensional points wouldn't think they are a vector space. In fact, many times in higher level math(s), viewing ℝn as a vector space is detrimental, since it really isn't one unless you make it. I know it may sound confusing right now, but it'll make sense as I explain it.

Let's frame it like this: do you know what "cat" plus "dog" equals? Do you know what 2 times "dog" equals? No, of course you don't. But if I told you that n*string* = *that string repeated n times*, and that *string1* + *string2* = *string1 string2*, you would know what each of those would be. Just as in early algebra you had no clue that (1, 2) + (2, 1) = (3, 3), or that 2(1, 2) = (2, 4). This is because ℝ2 was equipped with those properties by us. It never came naturally, we gave it those properties so we could do math(s) with it. The set of all 2-d points doesn't come with an addition, we have to give it one, just as we had to give strings addition.

This is how we actually define a vector space: a set that has a defined scalar multiplication and addition, along with conforming to the vector space axioms. This probably doesn't make much sense. Let's frame it like this:

We have a set V (everything that is a property/element of the vector space is underlined), defined as V := {(a, b) | a, b ∈ ℝ}. This is our vector space! You might say, but it isn't true. You cannot add yet. Let's say adding is a function that inputs two elements of our set (an ordered pair of elements). We say that +((a, b), (c, d)) = (a + c, b + d) (the plus is underlined so that you know that it isn't real number addition, but instead vector addition). We denote this like this:

+V ⨯ V → V

V ∋ (v1, v2), (w1, w2), + : ( (v1, v2), (w1, w2)) ↦ (v1 + w1, v2 + w2)

This basically means that it maps from ordered pairs of elements in V to an element in V. We can do the same with scalar multiplication:

· : ℝ ⨯ V → V

a ∈ ℝ, (v1, v2) ∈ V, (a, (v1, v2)) ↦ (av1, av2)

We then say this is the vector space (V+·), and it's elements are vectors. This is only here since we have defined vector addition and scalar multiplication. Why go through this trouble? Well, did you know that polynomials can be vectors given a vector space?

We define P as P := {p | p(x) = n=0N pnxn ∀N ∈ ℕ}, where (p + q)(x) = p(x) + q(x), (λ · p)(x) = λp(x).

We say that (P+·) is a vector space, even though it isn't a collection of numbers and isn't a subset of ℝn.

My challenge to you is to find some other examples of vector spaces that aren't just collections of numbers. Good luck!

That one guy that's always on

 
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I never even included the axioms to make it more simple and less math-y.

That one guy that's always on

 
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A point becomes definite if the  n   real numbers are given as its coordinates. The point is a math object that carries identity. The multitude of all points constitute a space of points or space of independent variables. All other math objects (scalars, vectors, tensors,...) are fields in this space. This space definitely not a vector space. A transformation of coordinates consists of arbitrary functions (inverse transformation is required). A transformation of a vector uses the partial derivatives of these functions. The points and the vectors are the different math objects. I've never seen any fields  in a vector space! Every math object (point, scalar, vector, tensor, ...) has its definition and the transformation law. Therefore, any math object carries its identity through arbitrary transformation of coordinates.

I was going for a very simplified version of what everything is. That's why I never included the axioms. I know I probably didn't use a lot of the correct terminology, but it was for a simplistic explanation that would be understood by people who haven't learned much high-degree mathematics. Technically a vector space consists of points, yes, but I was referring to the points in Euclidean space (who's only structure is a metric), which can be translated to the 2-dimensional Euclidean/real vector space (as they are ordered pairs of real numbers). Yes, the vector space is given an additional structure (namely scalar multiplication, vector addition, magnitude, and angle), but many people have the misconception that the only vectors are the ordered n-lets.

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The definition of Point and Vector must include their transformation laws. The Points and Vectors are math objects that carry their identity. A vector space consists of vectors (not of points). It is still a question if there exist a scalar distance between 2 close vectors. The metrics tensor exists not in the vector space but in the space of points (space of locations). All the fields depend of coordinates of points (not of the components of vectors). Important:  we can not assign algebra to undefined objects (expecting that assigned algebra will define the objects in the end). First we have to define the objects, after that the possible algebra follows automatically. Dot product, vector addition, magnitude, and vector multiplication are the consequences of the transformation laws (not our will). 

I never explicitly defined what vectors are, just vector spaces. I am going for the approach to eliminate the misconceptions that the only vectors and vector spaces are the ones who are/made from ordered collections of real numbers. Nothing about going into the vector transformation laws or any other laws. I just want people to see that you can make vector spaces that aren't subsets of ℝn. Nice to see criticism of my explanations, so I will try and make a post that is more mathematically rigorous, if that is what you want.

P.S., under the definition of a point, an element of a space (set that has added structure), a vector is technically a point (in the mathematical sense).

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Keep in mind that we are using different meanings behind the same words (I have no idea what do you mean by the word structure). It is very bad and I don't know how to jump out of this. In particular the words Point, Vector, math object. The term "math object" - is something unique, something that can carry unique identity, something that I can communicate to you and it will conserve its identity. Example:  real number  2.561 (or any definite real number). From the Point or Vector I require the same. In the coordinate system  S  I can assign the definite coordinates to the Point and the definite components to the Vector and they become unique. But I have to know the coordinates of the mentioned Point and the components of the mentioned Vector in the new coordinate system  S'  which is obtained by the given arbitrary transformation of coordinates. So we have to know the transformation laws for Point and for Vector. These laws are different. So we can add Point and Point and  Vector and Vector but we can not add Point and Vector. So n-tuple without transformation laws is nothing!  I took the words Point, Vector, math object, put a new meaning into them and I claim that you have to give up these words to me.  The my use of these words historically justified. 

I am sorry. We come from different backgrounds. I learned the mathematical definition of a vector (an element of a vector-space), while you learned the physics definition (an object with components, of which follow certain transformation rules). We have been arguing over something that has different context depending on your background, so I am sorry for the confusion.

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You said:   "you had no clue that (1, 2) + (2, 1) = (3, 3), or that 2(1, 2) = (2, 4). This is because ℝ2 was equipped with those properties by us."     My notations are different. We have vector   A(k)  and vector  B(l).  In 2-d space  k, l   can be 1 or 2. No abstract algebra,  A(1), A(2), B(1), B(2)  are real numbers. Because they are real numbers we can add them:   A(k)+B(l) . But because of the transformation law of a vector the adding A and B with different indexes  k and  l  has no meaning. So it has to be:   A(k)+B(k) . This obviously will be another vector  call it  C(k).  In your case  A(1)=1,  A(2)=2, B(1)=2, B(2)=1,  so  C(1)=3,  C(2)=3. The same way can be figured the multiplication of a scalar on a vector.  So I have a clue!  I used only transformation of a vector and the algebra of real numbers. 

I said in "early algebra", meaning when you were plotting point on a graph and solving linear equations. And I already said, we have different backgrounds and definitions. If you look at this lecture, for example, it defines vectors using vector-spaces and gives the polynomial example. You can also see it in the wikipedia article. If you want to know more about the mathematical, and not physics-based, definitions, you can look at these videos:

https://www.youtube.com/watch?v=xhTciBubSfM

https://www.youtube.com/watch?v=ErmR8W8KKUY

https://www.youtube.com/watch?v=y5fsXd1PPAI

The mathematical and the physics definitions for vector and vector space are no different learning them here in the UK. It is true that when we first learn vectors in physics (and maths) they are overly simplified as basically just arrows. I don't understand all this confusion!

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Yes, I agree with you, the official definition of vector is nonsense as it was long ago and up to now. Arrow, amplitude, direction - nonsense! But we are different in how to fix the situation. You are claiming that let us begin with vector space. But I am claiming that we need to fix  the definition of the vector itself by adding its transformation law and proclaiming that vector is "identity carrier" . It is a new kind of number - higher than the real number itself. The algebra of vectors will follow after the vector becomes unique. Only after we have unique single vector we can talk about "vector space" not in reverse (!).  Please, try to prove your position

What is  Rn ? Vector space? Claim 1 : in mathematics one can not postulate a space without having a definite elements of that space. You claim that your elements are vectors and they are definite because you assigned some algebra to your vectors.  I claim that algebra can not be assigned, it has to follow from the definition of a vector as a unique math object (by this I am in a conflict with the abstract algebra). Actually we need the space of Points (space of locations), We do not need a Vector space. A translation changes the coordinates of a Point but it does not change the components of a Vector.  Points and Vectors are different math objects.  Rn  is actually not a vector space - it is the space of Points (or space of independent variables).               

I have defined a vector as an element of a vector-space. Every component of the vector-space has to obey by the vector-space axioms, where the algebra and operations are defined to follow them. This is discussed here, and is very different depending on your major. Please, stop arguing. The definitions we've been given are different, so there is no need complaining over this. Thank you, but for mathematicians, this post is mostly correct. 

P.S. We do have a structure behind these (the axioms), it just isn't as specific as physicists like it to be.

I am sorry for all of this confusion. I will make a post explaining the physicist way, as well. This is the method of abstract algebra (defining a set with properties and operations, then defining the elements (ex groups), while yours is the way of practical real-world applications). I hope you understand this, and open your mind to this definition, as it is a common definition you will see later on.

Thanks for the critiques, but we are both right in the end!

I also do define the elements of the Vector Space, but they have to apply to the axioms. It follows the rules, it's just a different perspective. If you want to make your own post explaining it, go right on ahead. I will definitely read it. If you don't want to, I can make that post as best I can.

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