I'm trying to understand the following proof:
Given k points $p_1,...,p_k$ in $\mathbb{R}^n$. Then (for all $i,j =1,...,k$):
$$p_i+\text{span}(p_1-p_i,...,p_k-p_i)=p_j+\text{span}(p_1-p_j,...,p_k-p_j)$$
The proof starts like this:
Since $p_l-p_j=(p_l-p_i)-(p_j-p_i)$ for all $l,i,j=1,...,k$, it's true that $\text{span}(p_1-p_i,...,p_k-p_i)=\text{span}(p_1-p_j,...,p_k-p_j)$
How do you get to this conclusion? I understand that in this context $l$ varies and $i,j$ are set so on the left side of the equation there's a certain vector for each $l$ and on the right side you have a different vector for each $l$ plus a set vector that's the same for each $l$. But I just don't see why you get to this conclusion.
Let's call $v_l=p_l-p_i$ and $w_l=p_l-p_j$. The equation $$ p_l-p_j=(p_l-p_i)-(p_j-p_i)\tag1 $$ just means that $w_l=v_l-v_j$ which gives $$ \text{span}\{w_k\}\subset\text{span}\{v_k\}.\tag2 $$ On the other hand, the equation $(1)$ is $w_l=v_l+w_i$, i.e. $v_l=w_l-w_i$ which gives $$ \text{span}\{v_k\}\subset\text{span}\{w_k\}.\tag3 $$ Combine $(2)$ and $(3)$ to get the conclusion.