Change of basis transformation

change_basis.txt

Consider a standard coordinate system C in three dimensions. Let a linear transformation R be any rotation or reflection
around or through the origin respectively.

Let R be applied to the orthonormal basis vectors that define the axis directions of C. Let R be applied to any points
in C, say P₀ and P₁.

R: C → C'
R: P₀, P₁ → P'₀, P'₁

A new coordinate system C' results whose axis directions are the transformed axis directions of C under the
transformation R. The points P'₀, P'₁ will have the same coordinates in C' as they had in C since the transformed 
points did not move relative to the transformed basis vectors.

To distinguish between coinciding points in different coordinate systems let P'₀, P'₁ in C be designated Q₀, Q₁ in C'.
Also let P₀, P₁ in C be designated Q'₀, Q'₁ in C'.

Suppose we wish to discover the coordinates of the points P₀, P₁ in C', that is we wish to find Q'₀, Q'₁. 

Let R⁻¹ be the inverse of the transformation R. We note that R⁻¹ transforms P'₀, P'₁ to P₀, P₁ and therefore transforms 
Q₀, Q₁ to Q'₀, Q'₁. Since Q₀, Q₁ have the same coordinates in C' as P₀, P₁ in C, to find Q'₀, Q'₁ we simply need to 
apply R⁻¹ to P₀, P₁. Let P and Q' represent the same point but in C and C' respectively, then

R⁻¹: P → Q'                                                                      (Trivially, we can also say R: Q' → P)

The columns of the matrix R are the basis vectors of C' in C

let R = |a d g|  then  |1|   |1| |a|,  |0|   |0| |d|,  |0|   |0| |g|
        |b e h|        |0|→ R|0|=|b|   |1|→ R|1|=|e|   |0|→ R|0|=|h|
        |c f i|        |0|   |0| |c|   |0|   |0| |f|   |1|   |1| |i|

A matrix whose columns constitute an orthonormal basis is an orthogonal matrix. The inverse of an orthogonal matrix is
its transpose. That is

R⁻¹ = Rᵀ , and so

Rᵀ: P → Q'