# Question about points in general position (from Keith Conrad's expository paper on isometries)

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.

A hyperplane in $$\mathbb{R}^n$$ is the solution set of a single linear equation. Equivalently, it is the translation of a linear subspace of dimension $$n-1$$. For example, $$x - y = 1$$ defines a hyperplane in $$\mathbb{R}^2$$. It is a translation of the one-dimensional linear subspace given by $$x - y = 0$$.
Points $$P_0, \ldots, P_n$$ in $$\mathbb{R}^n$$ are in general position if there is no hyperplane that contains them all. If a collection of points is in general position, any translation of this set of points is in general position (do you see why?). Therefore, if $$P_0, P_1, \ldots, P_n$$ are in general position, then $$0, P_1 - P_0, \ldots, P_n - P_0$$ are in general position as well. Now if $$P_1 - P_0, \ldots, P_n - P_0$$ would be linearly dependent (in other words, if there is a non-trivial linear relation between them), they would be contained in an $$n-1$$-dimensional linear subspace. Since a linear subspace always contains $$0$$, this would be a contradiction.
• Is a hyperplane necessarily of dimension $n-1$? Can't a line in $\Bbb R^3$ be one? A translation of points in general position keeps them that way because the original set was already a translation of a subspace and a composition of two translation is a translation by the sum, making the outcome also points in general position, right? – V.Ch. Aug 17 at 22:50
• Yes, a hyperplane must be of dimension $n-1$. But note that if four points in $\mathbb{R}^3$ are contained in a line, they are also contained in a hyperplane (which in the case $n=3$ is just called a plane). Your idea about the translations is in the right direction, but be careful: a set of points in general position is not contained in a translation of a subspace. – merle Aug 17 at 23:16