Polytopes: Understanding Linear vs Affine Spans

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SUMMARY

The discussion clarifies the distinction between linear and affine spans using the points p=(1,0,0) and q=(0,1,0) in R3. The linear span of these points consists of all linear combinations, forming the plane defined by the points (0,0,0), (1,0,0), and (0,1,0). In contrast, the affine span includes all affine combinations, which require that the sum of coefficients equals 1, resulting in a line defined by the points (1,0,0) and (0,1,0). This fundamental difference is crucial for understanding polytopes and their geometric properties.

PREREQUISITES
  • Understanding of linear combinations and vector spaces
  • Familiarity with affine combinations and their properties
  • Basic knowledge of polytopes and their geometric representations
  • Proficiency in R3 coordinate system
NEXT STEPS
  • Study the concept of vector spaces in linear algebra
  • Learn about affine transformations and their applications
  • Explore the properties of polytopes and their classifications
  • Investigate the geometric interpretations of linear and affine spans
USEFUL FOR

Mathematicians, students of linear algebra, and anyone interested in the geometric properties of polytopes and their spans.

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 )
 
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??
 

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