Editing Orthonormal basis (section)

==Choice of basis as a choice of isomorphism==

For concreteness we discuss orthonormal bases for a real, <math>n</math>-dimensional vector space <math>V</math> with a positive definite symmetric bilinear form <math>\phi=\langle\cdot,\cdot\rangle</math>. 

One way to view an orthonormal basis with respect to <math>\phi</math> is as a set of vectors <math>\mathcal{B} = \{e_i\}</math>, which allow us to write <math>v = v^ie_i \ \ \forall \ v \in V</math> , and <math>v^i\in \mathbb{R}</math> or <math>(v^i) \in \mathbb{R}^n</math>. With respect to this basis, the components of <math>\phi</math> are particularly simple: <math>\phi(e_i,e_j) = \delta_{ij}</math> (where <math>\delta_{ij}</math> is the [[Kronecker delta]]).

We can now view the basis as a map <math>\psi_\mathcal{B}:V\rightarrow \mathbb{R}^n</math> which is an isomorphism of inner product spaces: to make this more explicit we can write
:<math>\psi_\mathcal{B}:(V,\phi)\rightarrow (\mathbb{R}^n,\delta_{ij}).</math>
Explicitly we can write <math>(\psi_\mathcal{B}(v))^i = e^i(v) = \phi(e_i,v)</math> where <math>e^i</math> is the dual basis element to <math>e_i</math>.

The inverse is a component map
:<math>C_\mathcal{B}:\mathbb{R}^n\rightarrow V, (v^i)\mapsto \sum_{i=1}^n v^ie_i.</math>
These definitions make it manifest that there is a bijection
:<math>\{\text{Space of orthogonal bases } \mathcal{B}\}\leftrightarrow \{\text{Space of isomorphisms }V\leftrightarrow \mathbb{R}^n\}.</math>

The space of isomorphisms admits actions of orthogonal groups at either the <math>V</math> side or the <math>\mathbb{R}^n</math> side. For concreteness we fix the isomorphisms to point in the direction <math>\mathbb{R}^n\rightarrow V</math>, and consider the space of such maps, <math>\text{Iso}(\mathbb{R}^n\rightarrow V)</math>. 

This space admits a left action by the group of isometries of <math>V</math>, that is, <math>R\in \text{GL}(V)</math> such that <math>\phi(\cdot,\cdot) = \phi(R\cdot,R\cdot)</math>, with the action given by composition: <math>R*C=R\circ C.</math>

This space also admits a right action by the group of isometries of <math>\mathbb{R}^n</math>, that is, <math>R_{ij} \in \text{O}(n)\subset \text{Mat}_{n\times n}(\mathbb{R})</math>, with the action again given by composition: <math>C*R_{ij} = C\circ R_{ij}</math>.