In an expository paper on isometries of $\Bbb R^n$ Keith Conrad proves the following corollary:
Corollary 2.7. Let $P_0,...,P_n$ be $n+1$ points in $\Bbb R^n$ in “general position”, i.e., they don’t all lie in a hyperplane. Two isometries of $\Bbb R^n$ that are equal at $P_0,...,P_n$ are the same.
So what exactly does "don't all lie in a hyperplane" mean? That these are $n+1$ linearly independent points? How is that possible in a vector space of dimension $n$? The final step is also something I have trouble with:
Upon subtracting $P_0$ from $P_0,...,P_n$, the points $0,P_1−P_0,...,P_n−P_0$ are in general position. That means no hyperplane can contain them all, so there is no nontrivial linear relation among $P_1−P_0,...,P_n−P_0$ (a nontrivial linear relation would place these $n$ points,along with $0$, in a common hyperplane), and thus $P_1−P_0,...,P_n−P_0$ is a basis of $\Bbb R^n$.
Why do the points remain "in general position" after one was substracted from the rest and why does that lead to their being a basis? If there were a nontrivial linear relation between these points does that mean that they'd be linearly dependent? I guess my issue is understanding the definition of "general position" and how it relates to linear dependence, so any help on that count is appreciated.