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