Affine Spaces of Rn

If $S\subseteq {\mathbb{R}}^n$ is a linear subspace and $b\in {\mathbb{R}}^n$ the set $b+S$ is called an affine subspace of ${\mathbb{R}}^n$ parallel to $S$. An affine subspace $b+S$ is a vector subspace iff $b\in S$. Wee see that $b+S=c+\tilde{S}$ iff $S=\tilde{S}$ and $b-\tilde{b}\in S$. Thus we can unambiguously define the dimension of $b+S$ to be dimension of $S$.

Suppose $v_0,\dots ,v_k$ are $k+1$ distinct points in ${\mathbb{R}}^n$. As long as $n\ge k$, then we know there's a $k$-dimensional affine subspace that contains such $k$-points. We say that the set $\{v_0,\dots ,v_k\}$ is affinely independent, or is in general position if it is not contained in any affine subspace of dimension strictly less than $k$.

Prop: For any $k+1$ distinct points $v_0,\dots ,v_k\in {\mathbb{R}}^n$, the following are equivalent.

• The set $\{v_0,\dots ,v_k\}$ is affinely independent.
• the set $\{v_1-v_0,\dots ,v_k-v_0\}$ is linearly independent.
• If $c_0,\dots ,c_k$ are real numbers such that $$\sum_{i = 0}^k c_i v_i = 0\quad \text{and} \quad \sum_{i = 0}^k c_i = 0,$$then $c_0=\cdots =c_k=0$.