Vector space

A vector space $V$ over a field $F$ is a set that satisfies certain axioms (see below) and which is equipped with two operations, vector addition and scalar multiplication. Vector addition is defined as a map

$+:V\times V\to V$

that takes the ordered pair $({\vec {u}},{\vec {v}})\in V\times V$ to the vector ${\vec {u}}+{\vec {v}}$. Here $\times$ represents the Cartesian product between sets. Scalar multiplication is defined in a similar way, as a map

$\cdot :F\times V\to V$

that takes the ordered pair $(a,{\vec {u}})\in F\times V$ to the vector $a\cdot {\vec {u}}$. Note that frequently the dot representing scalar multiplication is omitted, the result being written simply as $a{\vec {u}}$ instead. This is especially common when an inner product will also be defined on the vector space, with the dot then representing the inner product between two vectors. It is important to keep in mind the distinction between scalar multiplication, which multiplies one vector by a scalar, and an inner or scalar product, that combined two vectors to yield a scalar.

Axioms of a vector space

Let $V$ be a set, ${\vec {u}}$, ${\vec {v}}$, and ${\vec {w}}$ elements of that set, and $a$ and $b$ scalar elements of a field, $F$. Then $V$ is a vector space if the following axioms hold true for all choices of ${\vec {u}},\ {\vec {v}},\ a,\ b$

1. $V$ is closed under addition

The vector ${\vec {u}}+{\vec {v}}$ is also an element of $V$. This is automatically satisfied when the addition operation is defined as being injective as it was above. Care must be taken however if $V$ is a subset of some larger set $W$ and $+:V\times V\to W$, as is often the case when looking at subspaces.

2. Addition is commutative

The order in which two vectors are added does not affect the result, ${\vec {u}}+{\vec {v}}={\vec {v}}+{\vec {u}}$.

3. Addition is associative

${\vec {u}}+({\vec {v}}+{\vec {w}})=({\vec {u}}+{\vec {v}})+{\vec {w}}$. This means that even though addition is strictly defined as a binary operation, the object ${\vec {u}}+{\vec {v}}+{\vec {w}}$ is well defined.

4. An additive identity exists in $V$

Labeled ${\vec {0}}$, the additive identity or zero vector satisfies ${\vec {0}}+{\vec {u}}={\vec {u}}+{\vec {0}}={\vec {u}}$.

5. The additive inverse exists in $V$

A vector $-{\vec {u}}$ can be found such that $-{\vec {u}}+{\vec {u}}={\vec {u}}+(-{\vec {u}})={\vec {0}}$.

6. $V$ is closed under scalar multiplication

The vector $a{\vec {u}}$ is itself an element of $V$.

7. Scalar multiplication is distributive over addition in $F$

$(a+b){\vec {u}}=a{\vec {u}}+b{\vec {u}}$. It is important to note that the addition occurring on the left-hand side of this equality is a 'different operation' from the addition on the right-hand side. While the latter is vector addition as defined above, the former is the addition operation defined on the field $F$.

8. Vector addition is distributive over scalar multiplication

$a({\vec {u}}+{\vec {v}})=a{\vec {u}}+a{\vec {v}}$. In this case vector addition takes place on both sides of the equality.

9. Scalar multiplication is associative

$a(b{\vec {u}})=(ab){\vec {u}}$. This means that the algebraic structure of the underlying field $F$ is preserved. Note that the left-hand side of this equality contains two subsequent applications of the scalar multiplication defined above, while the right-hand side contains one scalar multiplication as defined in $F$ (that of $ab$), followed by scalar multiplication with the vector ${\vec {u}}$.

10. The multiplicative identity of $F$ provides a scalar multiplicative identity

$1{\vec {u}}={\vec {u}}$, where $1$ is the multiplicative identity of the field $F$.

These axioms can be expressed concisely in mathematical notation as follows:

$\forall {\vec {u}},{\vec {v}},{\vec {w}}\in V,\ \forall a,b\in F,$

1. ${\vec {u}}+{\vec {v}}\in V$
2. ${\vec {u}}+{\vec {v}}={\vec {v}}+{\vec {u}}$
3. ${\vec {u}}+({\vec {v}}+{\vec {w}})=({\vec {u}}+{\vec {v}})+{\vec {w}}$
4. $\exists {\vec {0}}\in V:{\vec {0}}+{\vec {u}}={\vec {u}}+{\vec {0}}={\vec {u}}$
5. $\exists \ -\!{\vec {u}}\in V:-{\vec {u}}+{\vec {u}}={\vec {u}}+(-{\vec {u}})={\vec {0}}$
6. $a{\vec {u}}\in V$
7. $(a+b){\vec {u}}=a{\vec {u}}+b{\vec {u}}$
8. $a({\vec {u}}+{\vec {v}})=a{\vec {u}}+a{\vec {v}}$
9. $a(b{\vec {u}})=(ab){\vec {u}}$
10. $1{\vec {u}}={\vec {u}}$