Editing Dual space (section)

=== Dimensional analysis ===
The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector <math>v \in V</math> can be paired with a covector <math>\varphi \in V^*</math> by the natural pairing
<math>\langle x, \varphi \rangle := \varphi (x) \in F</math> to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to [[Fraction#Reduction|reducing a fraction]]. Thus while the direct sum <math>V \oplus V^*</math> is a {{tmath|2n}}-dimensional space (if {{tmath|V}} is {{tmath|n}}-dimensional), {{tmath|V^*}} behaves as an {{tmath|(-n)}}-dimensional space, in the sense that its dimensions can be canceled against the dimensions of {{tmath|V}}. This is formalized by [[tensor contraction]].

This arises in physics via [[dimensional analysis]], where the dual space has inverse units.<ref>{{cite web
|url=https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/
|title=A mathematical formalisation of dimensional analysis
|last=Tao |first=Terence |author-link=Terence Tao
|date=2012-12-29
|quote=Similarly, one can define <math>V^{T^{-1}}</math> as the dual space to <math>V^T</math> ...
}}</ref> Under the natural pairing, these units cancel, and the resulting scalar value is [[dimensionless]], as expected. For example, in (continuous) [[Fourier analysis]], or more broadly [[time–frequency analysis]]:<ref group="nb">To be precise, continuous Fourier analysis studies the space of [[Functional (mathematics)|functionals]] with domain a vector space and the space of functionals on the dual vector space.</ref> given a one-dimensional vector space with a [[unit of time]] {{tmath|t}}, the dual space has units of [[frequency]]: occurrences ''per'' unit of time (units of {{tmath|1/t}}). For example, if time is measured in [[second]]s, the corresponding dual unit is the [[inverse second]]: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to <math>3s \cdot 2s^{-1} = 6</math>. Similarly, if the primal space measures length, the dual space measures [[inverse length]].