Some questions on the definition of $n$-simplex. Hatcher P102 last paragraph:

$n$-simplex is the smallest convex set in $\mathbb{R}^m$ containing $n+1$ points $v_0, \dots, v_n$ that do not lie in a hyperplane of dimension less than $n$.

I found this statement very confusing. By $\mathbb{R}^m$, does it just mean a space greater or equal to $n$? And also, by "do not lie in a hyperplane of dimension less than $n$", does it mean these $n$ points form $n$ linearly independent vectors, which lies in exactly a hyperplane of dimension $n$?
http://www.math.cornell.edu/~hatcher/AT/ATch2.pdf

For your question, $\mathbb{R}^{m}$ is the ambient space. An $n$ -simplex is a subset of $\mathbb{R}^{m}$ linearly homeomorphic to the set
$$\{x_{0}+x_{1}\cdots+x_{n}\}= 1,x_{i}\ge 0,0\le i\le n$$
You should think about what this is like in $\mathbb{R}^{n}$, and how it might like in $\mathbb{R}^{m},m\ge n$ in general. Draw some pictures might help. Beaware that there are $n+1$ points, not $n$ points, for a $n$-simplex. They are not in a hyperlane of dimension less than $n$. 
