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August 3rd, 2017, 09:13 PM   #1
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Affine space

Hello everybody,

I fail to grasp the idea of "affine space". I find different definitions in the web, but I can't understand them.

Parallel lines are the fundamental concept in Affine geometry; parallel lines are not dependent on distance and angle. So far, so good. We have clear definition for affine geometry: it is the study of parallel lines.

But what is "affine space"... explained for a layman? Can you give me, please, a clear definition? I need a good intuitive understanding of the basic concept of "affine space"....

Thank you!!
Have a nice day!!!!
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August 3rd, 2017, 11:55 PM   #2
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Math Focus: Yet to find out.
Do you know what we mean by 'space' when we say things like 'vector space' or 'Euclidean space'?
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August 4th, 2017, 12:51 AM   #3
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Quote:
Originally Posted by Joppy View Post
Do you know what we mean by 'space' when we say things like 'vector space' or 'Euclidean space'?
I know the basics of Euclidean space...
So...
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August 4th, 2017, 03:43 AM   #4
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Given a vector space a "subspace" is any subset of vectors that is closed under addition of scalar multiplication. In particular, if u and v are vectors in the subspace and a and b are scalars then au+ bv is also in the subspace. In particular, taking v= u, a= 1, b= -1, u- u= 0 is in the subspace. Geometrically, that can be visualized as a line or plane containing the origin.

An "affine" space can be visualized as a line or plane (or higher dimensional hyper-plane) that does NOT contain the origin. In general, if u and v are vectors in an affine space, u+ v is NOT in the space nor are au and bv. But any such affine space is parallel to a subspace. In particular, choosing any one vector, w, in the affine space, for any u in the affine space, u- w is in the corresponding subspace. That defines a one-to-one correspondence between the affine space and the subspace. We can define a "sort" of addition and scalar multiplication by "if u and v are in the affine space then u- w and v- w are in the subspace so that a(u- w)+ b(v- w)= au+ bv- (a+ b)w is in the subspace and so au+ bv- (a+ b)w+ w= au+ bv- (a+ b- 1)w is in the affine space.

For example, the plane 2x+ y= 3 does not contain the origin but is parallel to the plane 2x+ y= 0 that does. We can take w= <0, 3> as a vector in that affine space. To add u= < 1, 1> and v= <2, -1>, vectors in that affine space, we take u- w= <1, -2> and v- w= <2, -4>, both of which satisfy 2x+ y= 0, and add them: <1, -2>+ <2, -4>= <3, -6>, again in 2x+ y. Now go back to 2x+ y= 3 by adding <0, 3>: <3, -6>+ <0, 3>= <3, -3>. We define the addition of u= <1, 1> and v= <2, -1>, in this sense, to be <3, -3>.
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August 6th, 2017, 01:13 AM   #5
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You posted this in geometry, but the answers so far have been algebraic.

This is because (some) modern mathematicians have reduced geometry to algebra.

However, the geometrical point of view still also has merit.

I'm sorry, I'm in the north of Scotland at the moment so can't do sketches which would make this easier.

Anyway, affine spaces are important in modern Physics because they generalise transformations in the same way as follows for a simple straight line in a plane.

A straight line through the origin (do you know what this is?) has the equation y = mx where m is the slope.

You should do this next exercise for yourself.


Draw some x-y axes.
Draw in a series of straight lines through the origin, all with different m.
(No numbers are needed)

You should be able to easily see that any one of these lines can be transformed into any other one simply by rotating it about the origin.

This is what is meant by a linear space.
It is the geometry of rotation.

But there are other straight lines, which cannot be transformed into one of ours.
None of these pass through the origin and none have that simple equation.

All these extra lines have the equation y = mx + c, where c is a nonzero constant.

In junior high school, these equations are often called linear graphs because they refer to a straight line.

So mark on your sketch any point c, on the y axis.
Now draw a straight line through it, parallel to one of your earlier lines through the origin.

You should be able to see that it is impossible to transform the new line through c into any of the original lines without moving it (to pass through the origin).

Moving it is called translation.
This is a second type of transformation.

So to perform the overall transformation of y = nx into y = mx + c we require 2 transformations, a rotation and a translation.
These transformations are performed one after the other.

y = mx + c is an example of an affine space.

To recap, this is very important because it generalises transformations on more complicated objects than points on a plane from linear transformations to affine transformations.

Last edited by skipjack; August 6th, 2017 at 03:05 PM.
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