A calculus of the absurd
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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:
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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,
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1. The first four axioms are equivalent to stating that \((\textsf {V}, +)\) must be an Abelian group.
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2. We have two kinds of distributivity, one is that if \(\mathbf {x}, \mathbf {y} \in \textsf {V}\) and \(a \in \mathbb {K}\), then
\(\seteqnumber{0}{22.}{77}\)
\begin{equation}
a \cdot (\mathbf {x} + \mathbf {y}) = a \cdot \mathbf {x} + a \cdot \mathbf {y}
\end{equation}
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3. The second is that if \(a, b \in \mathbb {K}\) and \(\mathbf {x} \in \textsf {V}\), then
\(\seteqnumber{0}{22.}{78}\)
\begin{equation}
\mathbf {x}\cdot (a + b) = \mathbf {x}\cdot a + \mathbf {x} \cdot b
\end{equation}
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4. The neutral element (e.g. in \(\mathbb {R}\) this is \(1\)) in \(\mathbf {K}\) has the following property,
\(\seteqnumber{0}{22.}{79}\)
\begin{equation}
\forall \mathbf {x} \in \textsf {V} \hspace {12pt} 1 \cdot \mathbf {}{x} = \mathbf {x}
\end{equation}
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5. We also have a kind of “multiplicative distributivity”
\(\seteqnumber{0}{22.}{80}\)
\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.