2024-05-05

565: Subset of Affine-Independent Set of Points on Real Vectors Space Is Affine-Independent

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description/proof of that subset of affine-independent set of points on real vectors space is affine-independent

Topics


About: vectors space

The table of contents of this article

Starting Context

Target Context


  • The reader will have a description and a proof of the proposition that any subset of any affine-independent set of points on any real vectors space is affine-independent.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description


Here is the rules of Structured Description.

Entities:
V: ∈{ the real vectors spaces }
{p0,...,pn}: ⊆V, ∈{ the affine-independent sets of points on V}
{p0′,...,pm′}: ⊆{p0,...,pn}
//

Statements:
{p0′,...,pm′}∈{ the affine-independent sets of points on V}.
//

2: Natural Language Description


For any real vectors space, V, and any affine-independent set of points on, V, {p0,...,pn}, any subset, {p0′,...,pm′}⊆{p0,...,pn}, is affine-independent.


3: Proof


p0′=pk for a k∈{0,...,n}.

{p0−pk,...,pk−pk^,...,pn−pk} is linearly independent, where the hat mark denotes that the element is missing.

{p1′−p0′,...,pm′−p0′} is a subset of {p0−pk,...,pk−pk^,...,pn−pk}, and so, is linearly independent, so, {p0′,...,pm′} is affine-independent.

References
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