21.2: Vector Spaces

Vector spaces are an abstract concept used in math. So far we have talked about vectors of real numbers ($R^n$). However, there are other types of vectors as well. A vector space is a formal definition. If you can define a concept as a vector space then you can use the tools of linear algebra to work with those concepts.

A Vector Space is a set $V$ of elements called vectors, having operations of addition and scalar multiplication defined on it that satisfy the following conditions ($u$, $v$, and $w$ are arbitrary elements of $V$, and $c$ and $d$ are scalars.)

Closure Axioms

1. The sum $u+v$ exists and is an element of $V$. ($V$ is closed under addition.)
2. $cu$ is an element of $V$. ($V$ is closed under multiplication.)

Addition Axioms

1. $u+v=v+u$ (commutative property)
2. $u+(v+w)=(u+v)+w$ (associative property)
3. There exists an element of $V$, called a zero vector, denoted 0, such that $u+0=u$
4. For every element $u$ of $V$, there exists an element called a negative of $u$, denoted $−u$, such that $u+(−u)=0$.

Scalar Multiplication Axioms

1. $c(u+v)=cu+cv$
2. $(c+d)u=cu+du$
3. $c(du)=(cd)u$
4. $1u=u$