What exactly does it mean for two points to be barycentrically independent?

In summary, the conversation discusses the definition of barycentric independence in a vector space V, which states that a set of points {p0, p1, ..., pk} are barycentrically independent if the vectors formed by subtracting p0 from each of the other points are linearly independent. The issue arises with the definition for two points, as it only works for k≥2. However, it is clarified that two distinct points are automatically barycentrically independent and the vector they form must be nonzero.
  • #1
Mathguy15
68
0
Hello, I've been studying some linear algebra, and i am stuck on a certain definition.
The book i am using says that points{p0,p1,...,pk} in a vector space V are barycentrically independent if and only if the vectors {p1-p0,p2-p0,...,pk-p0} are linearly independent. The problem i have is the definition for two points. The definition in the book seems to work only for k≥2, but what about k=1? I'm sorry if I am missing something completely obvious, and i'll check more carefully if this is the case.



mathguy15
 
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  • #2
Two (distinct) points are automatically barycentrically independent, in the same way that a single (nonzero) point in automatically linearly independent
 
  • #3
so two points are barycentrically independent if and only if the vector they form is nonzero?
 
  • #5
Thanks!
 

1. What is barycentric independence?

Barycentric independence refers to the geometric concept that two points are not dependent on each other in terms of their position within a given shape or system. This means that the location of one point does not affect the location of the other point.

2. How is barycentric independence determined?

Barycentric independence is determined by analyzing the ratios of the distances between the points and the vertices of a shape. If the ratios are different for each point, then the points are considered to be barycentrically independent.

3. What is the significance of barycentric independence?

Barycentric independence is important in various fields, such as mathematics, physics, and computer graphics. It allows for the accurate representation and manipulation of shapes and systems, and it also has applications in areas such as optimization and data analysis.

4. Can more than two points be barycentrically independent?

Yes, more than two points can be barycentrically independent. In fact, the concept can be extended to any number of points within a given shape or system. However, the determination of barycentric independence becomes more complex as the number of points increases.

5. What is the difference between barycentric independence and collinearity?

Collinearity refers to the geometric concept of points lying on the same line. Barycentric independence, on the other hand, refers to the independence of points in terms of their position within a shape or system. Two points can be collinear but not barycentrically independent, and vice versa.

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