Polytopes: Understanding Linear vs Affine Spans

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In summary, the introductory material on polytopes discusses the points p=(1,0,0) and q=(0,1,0) in R3 and points out that their linear span and affine span are not the same. The linear span is the set of all linear combinations of p and q, while the affine span is the set of all affine linear combinations, requiring the sum of coefficients to be 1. This results in the linear span being a plane and the affine span being a line. This difference is due to the requirement of a sum of coefficients in an affine linear combination.
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arthurhenry
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The context is polytopes...reading some introductory material.

It talks about two points in R3, namely p=(1,0,0) and q=(0,1,0)
and tells me to notice that the linear span of of p and q and the affine span of p and q are not the same.

Could somebody tell me the difference? Thanks
 
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My guess is that the linear span is the set of all linear combinations of p and q and the affine span is the set of all affine linear combinations of p and q. (An affine linear combination requires that the sum of coefficients in the linear combination is 1 )
 
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The linear span will be the plane <(0,0,0),(1,0,0),(0,1,0)>.
The affine span will be the line <(1,0,0),(0,1,0)>.

Do you see why??
 

What are polytopes?

Polytopes are geometric objects in n-dimensional space that are bounded by flat surfaces, such as polygons in 2D or polyhedrons in 3D. They can also be called n-polytopes or n-polyhedra.

What is the difference between linear and affine spans?

The linear span of a set of vectors is the set of all possible linear combinations of those vectors. The affine span, on the other hand, includes translations and is therefore a larger space. In other words, the affine span is the linear span plus any possible translations.

Why is understanding linear vs affine spans important in the study of polytopes?

Polytopes can be described and studied using their vertices and edges, which can be represented as vectors. Understanding linear and affine spans allows us to determine the dimension and properties of the polytope, such as whether it is convex or non-convex.

How are linear and affine spans related to the concept of convexity?

A polytope is convex if and only if its affine span is equal to its linear span. This means that for a polytope to be convex, it must have no translations in its affine span, or in other words, it must be contained in its linear span.

Can a polytope have more than one linear span?

Yes, a polytope can have multiple linear spans. This is because the linear span of a set of vectors depends on which vectors are chosen as the basis. However, the affine span will always remain the same for a given polytope.

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