In a course called "Topics in Geometry and Topology", my teacher showed this in the case $k_i\in\mathbb{N}$ for all $i$ -- or rather, said he'd restrict himself to said case, but that the general case wasn't that much different. He said that these are affinely independent iff a certain matrix is nonsingular. Too bad I didn't write it down. The matrix should be:
$$\left(\begin{array}{ccc} k_1-k_0 & \dots & k_N-k_0 \\ \vdots & \ddots & \vdots \\ k_1^N-k_0^N & \dots & k_N^N-k_0^N \end{array}\right).$$
Or perhaps its transpose. Then he said something about subtracting the first column. So I did:
$$\left(\begin{array}{ccc} k_1-k_0 & \dots & k_N-k_1 \\ \vdots & \ddots & \vdots \\ k_1^N-k_0^N & \dots & k_N^N-k_1^N \end{array}\right).$$
Then he said something about expanding along line 1 and obtaining a Vandermonde determinant. And I'm like: wait a sec, Vandermonde has powers, not differences of powers! I have no reason to believe adding $(k_0,\dotsc,k_0^N)$ would leave the determinant unchanged as the passage above did. And I completely fail to see how Vandermonde matrices appear here. So how do I go about proving this has nonzero determinant?
Extra
In the book Simplicial Structures in topology, which he co-authored, my teacher has the following proof:
I am under the impression this needs to assume $i_0=0$, and I also can't quite get how he got the first equation, since there is no coordinate equal to 1… besides, $2x+1$ coordinates should give $2n+1$ equations, and there are $2n+2$ there… any ideas how to solve these problems?
Said teacher here.
The statement and proof of the theorem are ugly brief approximations. FIY, it was 2015-10-19 (in-class) when the proof was given. Same notation: $P^k = (k,k^2,\ldots, k^N)$.
Let $k_0$, $k_1$, $\ldots$ , $k_N$ be integers (distinct numbers), then the corresponding $P^{k_0}$, $P^{k_1}$, $\ldots$, $P^{k_N}$ are (affinely) linearly independent if and only if the $N+1$ vectors in homogeneous coordinates $$[1,k_j,k_j^2,\ldots, k_j^N]$$ are lin.ind.
Then Vandermonde matrixness occurs.
The formulation in the screenshot-book is kind of obscure, but trying (and failing) to state the same fact. Difference between affine and vector space.