N-plane

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This requires some pictures or at least discussion, the constraint that [ilmath]\sum^N_{i=0}t_i=1[/ilmath] is interesting. It doesn't require that they're positive so things like [ilmath]t_0=100[/ilmath] and [ilmath]t_1=-99[/ilmath] are possible.

Note: Capital-[ilmath]N[/ilmath] is used on this page because the [ilmath]N[/ilmath]-plane is constructed from a set of [ilmath]N+1[/ilmath] points - of [ilmath]\mathbb{R}^n[/ilmath] - be careful with this distinction.

Definition

Given a geometrically independent set of points of [ilmath]\mathbb{R}^n[/ilmath], [ilmath]\{a_0,\ldots,a_N\} [/ilmath] we define the [ilmath]N[/ilmath]-plane, [ilmath]P[/ilmath] spanned by these points to be the set:

• [math]\left\{x\in\mathbb{R}^n\ \left\vert\ x=\sum^N_{i=0}t_ia_i\wedge \sum_{i=0}^Nt_i=1\text{ for }t_i\in\mathbb{R}\right\}\right.[/math]

It can also be described as the set of points:

• [math]\left\{x\in\mathbb{R}^n\ \left\vert\ x=a_0+\sum^N_{i=1}t_i(a_i-a_0)\text{ for }t_i\in\mathbb{R}\right\}\right.[/math]

Notes for development of the page

Clearly from the second definition the plane is infinite in size, so the sum of t being 1 constraint must play some part in keeping it going through [ilmath]a_0[/ilmath]. I also need to prove the claim that these are indeed equivalent.