# A calculus of the absurd

#### 20.4 Vector spaces

##### 20.4.1 The vector space axioms

There are eight axioms in total, but I find it easier to remember them this way:

• Definition 20.4.1 A vector space is a set $$\textsf {V}$$ over a field $$\mathbb {K}$$ (elements of which are called “scalars”) equipped with an operator $$+$$ called “vector addition” which is an operator taking two elements of $$V$$ and returning a single element in $$V$$, and an operator $$\cdot$$ called “scalar multiplication” which takes a scalar and a vector, and outputs a vector.

We have the following eight axioms,

• 1. The first four axioms are equivalent to stating that $$(\textsf {V}, +)$$ must be an Abelian group.

• 2. We have two kinds of distributivity, one is that if $$\mathbf {x}, \mathbf {y} \in \textsf {V}$$ and $$a \in \mathbb {K}$$, then

\begin{equation} a \cdot (\mathbf {x} + \mathbf {y}) = a \cdot \mathbf {x} + a \cdot \mathbf {y} \end{equation}

• 3. The second is that if $$a, b \in \mathbb {K}$$ and $$\mathbf {x} \in \textsf {V}$$, then

\begin{equation} \mathbf {x}\cdot (a + b) = \mathbf {x}\cdot a + \mathbf {x} \cdot b \end{equation}

• 4. The neutral element (e.g. in $$\mathbb {R}$$ this is $$1$$) in $$\mathbf {K}$$ has the following property,

\begin{equation} \forall \mathbf {x} \in \textsf {V} \hspace {12pt} 1 \cdot \mathbf {}{x} = \mathbf {x} \end{equation}

• 5. We also have a kind of “multiplicative distributivity”

\begin{equation} a(b\mathbf {x}) = (ab) \mathbf {x} \end{equation}

Not exactly the most exciting stuff, but we can’t build castles without foundations! I’m not a structural engineer, but I’m pretty sure this is a true statement.