# Geometric independence

From Maths

**Stub grade: C**

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Suggest why this definition is useful

## Definition

Let us have a set of points, [ilmath]\{a_0,a_1,\ldots,a_N\}\subseteq\mathbb{R}^n [/ilmath]^{[Note 1]}, we say they are *geometrically independent* if^{[1]}:

- [math]\forall \{t_0,t_1,\ldots,t_N\}\subset\mathbb{R}\left[\left(\sum^N_{i=1}t_i=0\wedge\sum^N_{i=1}t_ia_i=0\right)\implies\left(t_0=t_1=\ldots=t_N=0\right)\right][/math]

The reader should note that this is very similar to linear independence; in fact, [ilmath]\{a_0,\ldots,a_N\} [/ilmath] is geometrically independent *iff* the set [ilmath]\{a_1-a_0,\ldots,a_N-a_0\} [/ilmath] is linearly independent

## Notes

- ↑ Consider [ilmath]n=0[/ilmath], then set equality is possible, hence [ilmath]\subseteq[/ilmath] rather than [ilmath]\subset[/ilmath] - see Importance of being pedantic about strict-subset and subset relations

## References

TODO: Is this in the right category?