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Tenshou

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P.S.

I think I know, yet what I know about this is incomplete.

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In summary, an affine space is a geometric space where points can be defined and measured, but there is no concept of adding points or multiplying them by a number as in a vector space. It can be thought of as a "flat" space, and can be transformed into a vector space by defining a coordinate system. Affine transformations can map any specific point to any point, while linear transformations must map the 0 vector to the 0 vector. An affine space is not a "special" type of vector space, and can be represented using euclidean space by forgetting about the origin. It is not required to be a topological space, but can be given a topology using a bijection from a vector space.

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Tenshou

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P.S.

I think I know, yet what I know about this is incomplete.

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HallsofIvy

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If you define a coordinate system on an affine space you can then define the 0 vector as given by the origin, other vectors as given by the coordinates of a point. So, basically, a "linear space" is an "affine space"

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Tenshou

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HallsofIvy

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Tenshou

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so an affine space is not a "special" type of vector space?

Edit: I mean what is the difference between an affine transform and a linear transform? besides they way they map points? is there any other difference in this?

Edit: I mean what is the difference between an affine transform and a linear transform? besides they way they map points? is there any other difference in this?

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HallsofIvy

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No, it is not. An affine space is not a vector space at all. Given an affine space, you can create a vector space from it by choosing any point in the affine space to be the "0" vector.Tenshou said:so an affine space is not a "special" type of vector space?

A linear transformation must map the 0 vector to the 0 vector. An affine transformation can map any specific point in the domain space to any point in the range space.Edit: I mean what is the difference between an affine transform and a linear transform? besides they way they map points? is there any other difference in this?

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mathwonk

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think of three dimensional euclidean (x,y,z) space. the x,y plane is a vector subspace because it has an origin. but some other plane not passing through the origin is just an affine subspace because it has no particular choice of origin.

a linear transformation of three space will take a plane through the origin to another plane through the origin, but an affine transformation may take it to a plane not through the origin.

i.e. even though euclidean space has an origin, an affine map does not see the origin, and treats

it as if there isn't one. in this way we can use euclidean space to represent affine space, by forgetting where the origin is.

linear transformations are made up of coordinate functions like this

ax+by+cz,

but affine transformations are made up of coordinate functions like this:

ax+by+cz + d.then we could say that the affine structure on euclidean space is that part of the euclidean structure that is preserved by affine transformations.

a linear transformation of three space will take a plane through the origin to another plane through the origin, but an affine transformation may take it to a plane not through the origin.

i.e. even though euclidean space has an origin, an affine map does not see the origin, and treats

it as if there isn't one. in this way we can use euclidean space to represent affine space, by forgetting where the origin is.

linear transformations are made up of coordinate functions like this

ax+by+cz,

but affine transformations are made up of coordinate functions like this:

ax+by+cz + d.then we could say that the affine structure on euclidean space is that part of the euclidean structure that is preserved by affine transformations.

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Tenshou

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Fredrik

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An affine space S isn't required to be a topological space, but you can always use the euclidean metric on the associated vector space V to define a topology on V, and then use it and a bijection from S into V to define a topology on S. (If f is such a bijection, there's a unique smallest topology on S such that ##f^{-1}(E)## is open for all open E). This way S will have all the topological properties of ##\mathbb R^n## for some n.Tenshou said:

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Fredrik

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I'm going to elaborate a bit on affine spaces. (I needed to get these ideas straight in my own head anyway).

Suppose that we want a set S to have the property that the*difference* between two arbitrary members of S is a vector, but other linear combinations as+bs' are undefined, and no member of S has an algebraic property that distinguishes it from the rest (like the 0 in a vector space).

We can use a map

$$\phi:V\times S\to S.$$ If this map has the property that for each ##(s,s')\in S\times S##, there's a unique ##v\in V## such that ##\phi(v,s)=s'##, then we can think of v as a "difference" s'-s. This interpretation is suggested by the notation ##\phi(v,s)=v+s##.

Suppose that we also want to be able to, for each s in S, define a vector space structure on S that makes s the additive identity (i.e. 0). Then we need to require some other things from ##\phi##. Suppose that ##\phi## has the property that for each s in S, the map ##\phi_s:V\to S## defined by ##\phi_s(v)=\phi(v,s)## for all v in V, is a bijection. Then we can try to use the map ##\phi_s## to define a vector space structure on V that makes s the additive identity (i.e. 0). Let ##a,b\in\mathbb R## and ##s',s''\in S## be arbitrary. We define the linear combination ##as'+bs''## by

$$as'+bs''=\phi_s\big(a\phi_s^{-1}(s')+b\phi_s^{-1}(s'')\big).$$ Does this make s the additive identity? No, not automatically.

$$s'+s=\phi_s\big(\phi_s^{-1}(s')+\phi_s^{-1}(s)\big).$$ Because of this, we also require that ##\phi(0,s)=0## for all s in S. This ensures that ##\phi_s^{-1}(s)=0##. And this simplifies the right-hand side above to s'.

This is all we need. Now we can call the triple ##(V,S,\phi)## an affine space. And then we can start abusing that terminology by calling S an affine space.

If you look up the definition of "affine space" in a math book, you will see that the conditions on ##\phi## are different, but the main reason why those conditions are used, is that they make the conditions I mentioned in this post true.

Suppose that we want a set S to have the property that the

We can use a map

$$\phi:V\times S\to S.$$ If this map has the property that for each ##(s,s')\in S\times S##, there's a unique ##v\in V## such that ##\phi(v,s)=s'##, then we can think of v as a "difference" s'-s. This interpretation is suggested by the notation ##\phi(v,s)=v+s##.

Suppose that we also want to be able to, for each s in S, define a vector space structure on S that makes s the additive identity (i.e. 0). Then we need to require some other things from ##\phi##. Suppose that ##\phi## has the property that for each s in S, the map ##\phi_s:V\to S## defined by ##\phi_s(v)=\phi(v,s)## for all v in V, is a bijection. Then we can try to use the map ##\phi_s## to define a vector space structure on V that makes s the additive identity (i.e. 0). Let ##a,b\in\mathbb R## and ##s',s''\in S## be arbitrary. We define the linear combination ##as'+bs''## by

$$as'+bs''=\phi_s\big(a\phi_s^{-1}(s')+b\phi_s^{-1}(s'')\big).$$ Does this make s the additive identity? No, not automatically.

$$s'+s=\phi_s\big(\phi_s^{-1}(s')+\phi_s^{-1}(s)\big).$$ Because of this, we also require that ##\phi(0,s)=0## for all s in S. This ensures that ##\phi_s^{-1}(s)=0##. And this simplifies the right-hand side above to s'.

This is all we need. Now we can call the triple ##(V,S,\phi)## an affine space. And then we can start abusing that terminology by calling S an affine space.

If you look up the definition of "affine space" in a math book, you will see that the conditions on ##\phi## are different, but the main reason why those conditions are used, is that they make the conditions I mentioned in this post true.

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Fredrik

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Fredrik

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You may need to be more specific. What about the notation bothers you?Tenshou said:I still don't understand the reason for the notation of the phi_s

We have a function ##\phi:V\times S\to S##, and we want to use it to define a function from V into S. How would you do it? The obvious way is to pick any point s in S, and then use the map ##v\mapsto\phi(s,v)##. It seems natural to call it ##\phi_s##, since I used the function ##\phi## and an ##s\in S## to define it.

(The words "function" and "map" are interchangeable in my posts, and in most books. There are however some authors who prefer to use "function" only when the codomain is ℝ or ℂ).

Why don't you post it here? If it's a serious math article, I don't think you will get in trouble with the moderators.Tenshou said:I recently found an article online which goes into a little more detail about this topic, If anyone would like they can message me, and then I can get around to sending them a link to the paper, it is rather interesting.

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Tenshou

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Fredrik said:You may need to be more specific. What about the notation bothers you?

We have a function ##\phi:V\times S\to S##, and we want to use it to define a function from V into S. How would you do it? The obvious way is to pick any point s in S, and then use the map ##v\mapsto\phi(s,v)##. It seems natural to call it ##\phi_s##, since I used the function ##\phi## and an ##s\in S## to define it.

Why don't you post it here? If it's a serious math article, I don't think you will get in trouble with the moderators.

It is from the University of Penn, I believe...

Affine Space

Why wouldn't you call it ##\phi_v## isn't that what the functor is acting on >.< I apologize if I am being a little picky. I just don't understand if the set of vectors or points are being acted on?

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Fredrik

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Functor? ##\phi_s## is just a function from V into S. A functor (in the category of vector spaces) would take vector spaces to vector spaces. ##\phi_s## takes vectors in a specific vector space V to points in a specific set S.Tenshou said:It is from the University of Penn, I believe...

Affine Space

Why wouldn't you call it ##\phi_v## isn't that what the functor is acting on >.< I apologize if I am being a little picky. I just don't understand if the set of vectors or points are being acted on?

For each s in S, there's a map ##v\mapsto \phi(s,v)## from V into S. I chose to denote it by ##\phi_s##. You're suggesting ##\phi_v##. First of all, how would you choose v? Second, these are infinitely many functions, one for each s, and you want to use the same notation ##\phi_v## for all of them? Then how would you interpret an expression like ##\phi_v(u)##, where u is a member of V?

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Tenshou

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Fredrik said:For each s in S, there's a map ##v\mapsto \phi(s,v)## from V into S. I chose to denote it by ##\phi_s##. You're suggesting ##\phi_v##. First of all, how would you choose v? Second, these are infinitely many functions, one for each s, and you want to use the same notation ##\phi_v## for all of them? Then how would you interpret an expression like ##\phi_v(u)##, where u is a member of V?

Okay never mind you answered my question. So, the elements ##s##, the point in the "affine set" are the one which are being "acted upon" by ##\phi_s##, correct? this is what I am confused about. 【・_・?】 Isn't the function ##\phi## a bi-linear "map"? or is it only a bi-linear where there is a tensor product between the two and not a Cartesian product?

##\phi:V\times S\to S## <----- This right here.

One moment, is ##\phi(s,v)## like a "scalar product" type thing? if so, then wouldn't that be like adding a vector with a point? how does this work?

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Fredrik

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No. I said that for each s in S, we define ##\phi_s:V\to S## by ##\phi_s(v)=\phi(s,v)## for all v in V. So each ##\phi_s## takes a member of V to a member of S.Tenshou said:So, the elements ##s##, the point in the "affine set" are the one which are being "acted upon" by ##\phi_s##, correct?

Suppose that X,Y,Z are vector spaces. A map ##f:X\times Y\to Z## is said to be bilinear if for all ##(x,y)\in X\times Y## both of the maps ##f_x:Y\to Z## and ##f_y:X\to Z## defined by ##f_x(v)=f(x,v)## for all v in Y and ##f_y(u)=f(u,y)## for all u in X, are linear.Tenshou said:this is what I am confused about. 【・_・?】 Isn't the function ##\phi## a bi-linear "map"? or is it only a bi-linear where there is a tensor product between the two and not a Cartesian product?

##\phi:V\times S\to S## <----- This right here.

When we define ##\phi##, we can't say that it's bilinear, since S isn't even a vector space. However, once we have used one of the ##\phi_s## maps to turn S into a vector space, then it makes sense to ask if ##\phi## is bilinear with respect to that vector space structure on S. It should be easy to check if it is, but I haven't done it myself, so I don't know for sure.

It doesn't work, unless weTenshou said:One moment, is ##\phi(s,v)## like a "scalar product" type thing? if so, then wouldn't that be like adding a vector with a point? how does this work?

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Tenshou

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Oh okay, thank for cleaning this up for me.

An affine space is a mathematical concept that refers to a geometric space where points are defined by their relative position to each other rather than by absolute coordinates. Essentially, it is a space that does not have a fixed origin or axes, and instead relies on the concept of displacement vectors to describe the relationships between points.

An affine space differs from a Euclidean space in that it does not have a fixed origin or axes. In a Euclidean space, the points are defined by their absolute coordinates in relation to a fixed origin, while in an affine space, the points are defined by their position relative to other points. Additionally, Euclidean spaces are limited to three dimensions, while affine spaces can exist in any number of dimensions.

Some examples of affine spaces include a plane, a line, and a flat surface in three-dimensional space. These spaces do not have a fixed origin or axes, and the relationships between points are described using displacement vectors.

Affine spaces play a crucial role in many areas of mathematics, including geometry, linear algebra, and functional analysis. They provide a framework for studying the relationships between points and can be used to solve complex geometric and algebraic problems. Additionally, affine spaces have applications in fields such as computer graphics, computer vision, and robotics.

An affine space is closely related to affine transformations, which are transformations that preserve the structure of an affine space. These transformations include translations, rotations, and reflections, and they are used to manipulate the relationships between points in an affine space. Affine transformations are essential in many fields, including computer graphics and computer vision.

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