$n$ points and $n-1$ dimensions

Question

2 points make a line -1D

3 points at most make a plane -2D

similary can we find $n$ points making up a $n-1$-dimensional object in $\mathbb{R}^n$?

Answer 1

Yes, under certain conditions. Let $p,q$ be points. Then denote by $v=p-q$ the vector from $q$ to $p$.

Let $p_0,...,p_{n-1}$ be $n$ point in $n$-space, make the $n-1$ vectors $v_1=p_1-p_0,...,v_{n-1}=p_{n-1}-p_0$. If this set of vectors is linearly independent, then the points uniquely define an $n-1$-dimensional hyperplane in $n$-space.