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)




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,wV: u+(v+w)=(u+v)+w
  • uV: u+0=u
  • uV uV: 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,vV: 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.



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


  • a? u,vV: a(u+v)=(au)+(av)
  • a,b? uV: (a+b)u=(au)+(bu)
  • a,b? uV: a(bu)=(ab)u
  • uV: 1?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.


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