2  Vector Spaces

Sometimes called linear spaces: vector spaces are linear by definition.

2.1 Definition of a Vector Space

A vector space \(\mathcal{V}\) over a field \(\mathcal{F}\) is a non-empty set \(V\) together with a binary operation

\[\oplus:V\times V\rightarrow V\,\,\text{(\emph{vector addition})}\]

and a binary function

\[\otimes:F\times V\rightarrow V\,\,\text{(\emph{scalar multiplication})}\]

[\(\oplus\), \(\otimes\) are my notation!] obeying the following axioms.

\(\forall|a\rangle,|b{\rangle},|c{\rangle}{\in}V;\,\,\alpha,\beta\in F\):

  • [V1]: \(|a\rangle\oplus|b\rangle\in V\) (closure under \(\oplus\))

  • [V2]: \(|a\rangle\oplus|b\rangle=|b\rangle\oplus|a\rangle\) (commutativity of \(\oplus\))

  • [V3]: \(\left(|a\rangle\oplus|b\rangle\right)\oplus|c\rangle=|a\rangle\oplus\left(|b\rangle\oplus|c\rangle\right)\) (associativity of \(\oplus\))

  • [V4]: \(\exists|0\rangle\in V||a\rangle\oplus|0\rangle=|a\rangle\) (identity element under \(\oplus\))

  • [V5]: \(\exists|-a\rangle\in V||a\rangle\oplus|-a\rangle=|0\rangle\) (inverse elements under \(\oplus\))

  • [V6]: \(\alpha\otimes|a\rangle\in V\) (closure under \(\otimes\))

  • [V7]: \(\alpha\otimes\left(\beta\otimes|a\rangle\right)=\left(\alpha\cdot\beta\right)\otimes|a\rangle\) (compatibility of \(\otimes\) with \(\cdot\))

  • [V8]: \(\exists1\in F|1\otimes|a\rangle=|a\rangle\) where \(1\) is the multiplicative identity in \(\mathcal{F}\) (identity element under \(\otimes\))

  • [V9]: \(\alpha\otimes\left(|a\rangle\oplus|b\rangle\right)=\alpha\otimes|a\rangle\oplus\alpha\otimes|b\rangle\) (distributivity of \(\otimes\) w.r.t. \(\oplus\))

  • [V10]: \(\left(\alpha+\beta\right)\otimes|a\rangle=\alpha\otimes|a\rangle\oplus\beta\otimes|a\rangle\) (distributivity of \(\otimes\) w.r.t. +).

Equivalently, a vector space is an ordered pair \(\mathcal{V}=\left(V,\left\{ \otimes,\oplus\right\} \right)\) obeying these axioms.

NB the zero vector \(|0\rangle\) should not be confused with the \(0\) element of the field.

V7 establishes that no ambiguity is introduced if we denote scalar multiplication \(\otimes\) with the same symbol \(\cdot\) used for field multiplication. Similarly, V10 establishes that \(\oplus\) can be written + without introducing any confusion. Strictly the two binary operations play different roles, but I have never seen them distinguished symbolically in the literature. Hence, from hereon I will use \(\cdot\) for both field multiplication and scalar multiplication (or will omit the symbol all together, as is customary), and \(+\) for both field addition and vector addition.

2.2 Examples of Vector Spaces

  • \(\left\{ |0\rangle\right\}\), the set containing only the zero vector

  • \(\mathcal{F}\), the field over which \(\mathcal{V}\) is defined (e.g. \(\left(\mathbb{R},\left\{ +,\cdot\right\} \right)\))

  • \(\mathcal{F}^{N}\), the \(N-\)dimensional co-ordinate space. This is perhaps the most familiar example. E.g.

\[|x\rangle=\left(\begin{array}{c} x_{1}\\ x_{2}\\ x_{3}\\ \vdots\\ x_{N} \end{array}\right);\quad|x\rangle+|y\rangle=\left(\begin{array}{c} x_{1}+y_{1}\\ x_{2}+y_{2}\\ x_{3}+y_{3}\\ \vdots\\ x_{N}+y_{N} \end{array}\right)\,\,\text{(\emph{etc}.)}\]

with the absolutely familiar case being \(\mathbb{R}^{N}\).