Axioms of a Vector Space

The following axioms must hold true for a Vector Space to be defined. Assume* $\vec{a},\vec{b},\vec{c}\in V$ and $\alpha ,\beta \in F$, where $V$ is the set of vectors and $F$ is the set of scalars.

V1. Associativity of Vector Addition §

We can change the parenthesis for vector addition, as long as the operands remain the same:$$\vec{a} + (\vec{b}+\vec{c}) = (\vec{a}+\vec{b})+\vec{c}

##### V2. Commutativity of Vector Addition The operands in vector addition can be flipped:$$\vec{a} +\vec{b} = \vec{b} +\vec{a}

V3. Existence of Zero Vector §

There must exist a vector $\vec{0}$ which does not affect vector addition on another vector:$$\vec{a} + \vec{0} = \vec{0} + \vec{a} = \vec{a}

##### V4. Existence of Additive Inverse For all vectors in $V$, there must exist an opposite vector, called the additive inverse. The sum of a vector and it's additive inverse must equal the zero vector:$$\vec{a} + \vec{a}' = \vec{0}

V5. Compatibility of scalar multiplication and field multiplication §

Scalar multiplication (scalar multiplied by a vector) and field multiplication ( scalar times a scalar) are compatible:

$$\alpha \times \underset{F\times V}{\underbrace{(\beta \times \vec{a})}}=\underset{F\times F}{\underbrace{(\alpha \times \beta )}}\times \vec{a}$$

Quick note

This one is usually called the associativity of scalar multiplication, but that’s wrong. When we do $\alpha \times \vec{a}$, it is very different from doing $\alpha \times \beta$, as one of them is scalar multiplied with a vector (scalar multiplication) , and another is a scalar multiplied with a scalar (field multiplication). By definition, a field has multiplication associativity with itself. Also note that this isn’t associativity of scalar multiplication. We don’t need $\alpha \times \vec{a}=\vec{a}\times \alpha$ to define a vector space.

V6. Multiplication of the unit scalar §

Scalar multiplication with the field unity does not affect the vector:

$$1_F\times \vec{a}=\vec{a}$$

Sometimes the field unity is called the unit scalar

V7. Distributivity of scalar multiplication with respect to vector addition §

We can expand out (distribute) the multiplication of a scalar to the sum of two vectors:

$$\alpha \times (\vec{a}+\vec{b})=(\alpha \times \vec{a})+(\alpha \times \vec{b})$$

V8. Distributivity of scalar multiplication with respect to field addition §

We can expand out (distribute) the multiplication of a sum of scalars to another vector:

$$(\alpha +\beta )\times \vec{a}=(\alpha \times \vec{v})+(\beta \times \vec{v})$$

Again here it’s important to note that associativity of multiplication doesn’t have to be defined. We can have a perfectly valid vector space with only left scalar multiplication being defined i.e. $\alpha \times \vec{a}$ without right multiplication ($\vec{a}\times \alpha$) being defined!