Explanation needed for the representation of simplex as a polyhedron

Question

In convex optimization book (by Stephen Boyd) it is mentioned that the simplex for $k+1$ points $v_0,\cdots v_k$ is defined as $$S=\{\theta_0v_0+\cdots\theta_kx_k| \theta_i\geq0, \sum_{i=0}^{k+1}\theta_i=1\}.$$ To show that they assume that the points are affinely independent. Based on this they form a matrix $B$ such that $$B=[v_1-v_0\quad v_2-v_0 \cdots v_k-v_0]$$ In the book author say that $B$(whose rows are $n$ and columns are $k$) has rank $k$. Does it means that for a certain dimensional space with $n$ dimensions the simplex can be defined for only $k+1\leq n$ number of points?

Answer 1

The definition of simplex assumes $\{v_0,...,v_k\}$ are affinely independent. That says that $k+1 \leq n$ since we cannot have more than $n$ independent elements in $\mathbb{R}^n$.