Équipollence

Of Latin Aequipollens , who has an equal value.

This term is a a little erudite synonym of the word equivalent, two sentences équipollentes are sentences having the same significance.

In secondary education, the vectors are usually defined as being classes of equivalence for the équipollence of the bipoints. The équipollence is then defined using some allowed intuitive geometrical concepts: Two bipoints (has, b) and (C, d) are équipollents if they have the same direction, the same direction and the same length.

In a space closely connected: Two bipoints (has, b) and (C, d) are known as équipollents if $\backslash \; overrightarrow\; \{has\; \backslash ,\; B\}\; \backslash ,\; =\; \backslash \; overrightarrow\; \{C\; \backslash ,\; D\}\; \backslash$, the points has, B, C, D form a parallelogram then (possibly flattened).

The équipollence can also be defined in a simple axiomatic way:

Équipollence. - Is X a nonempty unit whose elements will be called points. It is supposed that X×X is provided with a relation of noted equivalence ~ checking the two following axioms:

i) For all has, B, c∈X, it exists single a d∈X such as (has, b)~ (C, d).

II) For all (has, b), (a', b') ∈X×X, (has, b)~ (a', b') ⇒ (has, a') ~ (B, b').

The relation ~ will then be called a équipollence on X.

Crossing of the équipollences - If $\backslash \; overrightarrow\; \{has\; \backslash ,\; B\}\; \backslash ,\; =\; \backslash \; overrightarrow\; \{C\; \backslash ,\; D\}\; \backslash$, then $\backslash \; overrightarrow\; \{has\; \backslash ,\; C\}\; \backslash ,\; =\; \backslash \; overrightarrow\; \{B\; \backslash ,\; D\}\; \backslash$.

One from of deduced a new definition from spaces closely connected: Space closely connected. - a unit X provided with a équipollence is called a space closely connected. The classes of equivalence for the équipollence are called the vectors of X.

The class of a bipoint (has, b) will be noted $\backslash \; vec\; \{ab\}$.

The sum of two vectors we indicate by E the whole of the vectors of X. to define will need the following lemma:

Lemma. - If (has, b)~ (a', b') and (B, c)~ (b', it) then (has, c)~ (a', it). Proof. We have: (has, b)~ (a', b') ⇒ (has, a') ~ (B, b'), and (B, c)~ (b', it) ⇒ (B, b') ~ (C, it). Therefore, by transitivity, (has, a') ~ (C, it). From where (has, c)~ (a', it). Addition of the vectors. - Is $\backslash \; vec\; \{U\}$ and $\backslash \; vec\; \{v\}$ two vectors and has a point of X, it exists B, c∈X single such as: $\backslash \; vec\; \{U\}\; =\; \backslash \; vec\; \{ab\}$ and: $\backslash \; vec\; \{v\}\; =\; \backslash \; vec\; \{bc\}$. By definition $\backslash \; vec\; \{U\}\; +\; \backslash \; vec\; \{v\}\; =\; \backslash \; vec\; \{bc\}$.

The preceding lemma shows that this definition does not depend on point A. It is then easy to check that the unit E of the vectors of X provided with the addition is an abelian group.

So moreover E vector Space is a on a commutative body K one says that X is a Espace closely connected on the body K. If E is a Euclidean vector Space one says that X is a Espace closely connected Euclidean.