Mathematics is a formal precise language we have invented; It is essentially a short form and disciplined way of writing logical arguments. (18, July 2015)

Every mathematical object parametrized by n real numbers may be treated as a point of the **n-dimensional space** of all such objects. (Bernhard Riemann in 1854)

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Set

A **group** is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element. (C=A.B)

- ∀u,v,w∈V: u+(v+w)=(u+v)+w
- ∀u∈V: u+0=u
- ∀u∈V ∃−u∈V: u+(−u)=0

An** abelian group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. (commutativity, *AoB=BoA)*

- ∀u,v∈V: u+v=v+u

A** ring** is an abelian group with a *second binary operation that is associative (Ao(BoC)=(AoB)oC)*, is *distributive over the abelian group operation* (Ao(B.C)=(AoB).(AoC)) and* has an identity element*

A **field** is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication.

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A **vector space** is an abelian group of **vectors**, which may be multiplied (“scaled”) by numbers, called scalars in this context.

- ∀a∈𝕂 ∀u,v∈V: a⋅(u+v)=(a⋅u)+(a⋅v)

- ∀a,b∈𝕂 ∀u∈V: (a+b)⋅u=(a⋅u)+(b⋅u)
- ∀a,b∈𝕂 ∀u∈V: a⋅(b⋅u)=(ab)⋅u

- ∀u∈V: 1𝕂⋅u=u

A nonempty subset *W* of a vector space *V*** that is closed** under addition and scalar multiplication (and therefore contains the **0**-vector of *V*) is called a** subspace of V**.

Topology:

http://en.wikipedia.org/wiki/Simplex