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! Definitions: Basis of an affine subspace;

Consider a $d$-dimensional affine subset $D$ of an affine space, and a set of points $S$ in $D$.

  • If $S$ is a generating set for $D$, then $S$ contains at least $d+1$ points. A generating set with exactly $d+1$ points is a basis.

  • If $S$ is an independent set for $D$, then $S$ contains at most $d+1$ points. An independent set with exactly $d+1$ points is a basis.

  • If $|S| = d + 1$ and $S$ is not contained in a proper affine subspace of $D$, then $S$ is a basis.

Let's take the first bullet for example:

$\dim(D) = d$, meaning that $D$ is generated by $d+1$ (independent) points. Suppose that $S$ is a generating set for $D$. This means that the points in $S$ are not contained in a proper affine subspace of $D$. From this fact follows that we cannot find an affine subspace of $D$ that is generated by strictly less than $d+1$ points, which also holds for $S$. This results in $|S| \ge d+1$. I am struggling to find an explanation for the second part of the statement.

Is my attempt correct?

I'll give my proofs for the other two bullets too:

  • Suppose that $S$ is an independent set for $D$. By definition, for every point $x \in S$ there exists an affine subspace of $D$ that contains $S \backslash \{x\}$. Suppose that $|S| > d+1$ and take an arbitrary point $s \in S$. There has to be an affine subspace of $D$ that contains $S \backslash \{s\}$. However $|S \backslash \{s\}| > d$, meaning that there cannot be an affine subspace of $D$ (dimension $d$) that contains $S \backslash \{s\}$. This results in $|S| \le d+1$.

  • This one would follow directly from the definition of a generating set of an affine space and the first bullet.

Thank you!

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