A calculus of the absurd

22.3 Vector spaces

22.3.1 The vector space axioms

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

  • Definition 22.3.1 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.