# Topology/Vector Spaces

 Topology ← Perfect Map Vector Spaces Morphisms →

A vector space ${\displaystyle V}$ is formed by scalar multiples of vectors. The scalars are most commonly real numbers but can also be complex, or from any field.

## Definition

A vector space ${\displaystyle V}$  on a field ${\displaystyle F}$  is a set ${\displaystyle V}$  equipped with two binary operations: common vector addition on elements of ${\displaystyle V}$  and scalar multiplication, by elements of ${\displaystyle F}$  on elements of ${\displaystyle V}$ . These operations are subject to 8 axioms (u,v, and w are vectors in ${\displaystyle V}$  and a and b are scalars in ${\displaystyle F}$ ):

1. Associativity (addition): (u+v)+w = u+(v+w).

2. Associativity (scalar and field multiplication): a(bu) = (ab)u

3. Distributivity (field addition): (a+b)u = au+bu

4. Distributivity (vector addition): a(u+v) = au+av

5. Identity element (addition): ${\displaystyle \exists }$  0 ${\displaystyle \in V}$  such that u+0 = u ${\displaystyle \forall }$  u ${\displaystyle \in V}$

6. Identity element (scalar multiplication): 1u = u

7. Commutativity: u+v = v+u

8. Inverse element: ${\displaystyle \exists }$  -u ${\displaystyle \in V}$  such that u+(-u) = 0 ${\displaystyle \forall }$  u ${\displaystyle \in V}$

An example of basic arrow vectors: first the black vector as the sum of the red and blue vectors, then the black vector as a sum of the blue vector and a scaling of the red vector (x2)

 Topology ← Perfect Map Vector Spaces Morphisms →