Linear Combination of Points

Question

I am reading Lecture Notes on Discrete Geometry. It says "The linear span of a set $X$ can be described as the set of all linear combinations of points of $X$." Here $X\in{R^d}$. I am not sure what does it mean by linear combination of points. I only know the concept of linear combination of vectors. I haven't find any useful information by Googling these keywords. Anyone can help? Thanks!

Answer 1

A combination of a finite number of points $P_k$ written under the form :

$$P=\sum_{k=1}^n a_kP_k\tag{1}$$

defines a unique manner point $P$ (called the barycenter of points $P_k$ with weights $a_k$) under the condition that the $$\sum_{k=1}^n a_k=1\tag{2}$$

Two steps for giving an indisputable meaning to (1).

• first step: refer to the non-ambiguous notation with vectors for a given origin $O$ :

$$\vec{OP}=\sum_{k=1}^n a_k\vec{OP_k}\tag{3}$$

• second step: as being independent from the choice of origin $O$ (and this is where condition (2) is used) ; indeed, we can transform (3) in the following way :

  $$\color{red}{\vec{OO'}}+\vec{O'P}=\color{red}{\sum_{k=1}^n a_k\vec{OO'}}+\sum_{k=1}^n a_k\vec{O'P_k}\tag{4}$$

where the colored items, once cancelled, give a relationship analogous to (3), with $O$ replaced by $O'$.

Example :

$$G=\sum_{k=1}^n \dfrac1nP_k\tag{1}$$

is the center of gravity of the system of points $P_k$.