Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $  or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/oMjab.png

the maps

$$
f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$   $$
x\longmapsto (1,x,x^2,\cdots,x^{n-1})
$$
and
$$
f:\mathbb{R}^2=\mathbb{C}\longrightarrow \mathbb{R}\times\mathbb{C}^{n-1}=\mathbb{R}^{2(n-1)+1},
$$
$$
z\longmapsto (1,z,z^2,\cdots,z^{n-1})
$$
satisfy the condition:

***(C)*** ***for any distinct $x_1,x_2,\cdots,x_n$, their images are linearly independent**.* 

**Question:** how to construct maps satisfying **(C)** 

$$
f: \mathbb{R}^m\longrightarrow \mathbb{R}^{m(n-1)+1}
$$
for $m\geq 3$?