Change of basis

Let C be a right-handed three-dimensional coordinate system.

Let R be a linear transformation that preserves an orthonormal basis, such as a rotation around or reflection through the origin.

Let R be applied to any point in C, say P, and let Q be the image of P.

R : P ↦ Q

Let R be applied to the orthonormal basis vectors that define the axis directions of C to give a new coordinate system C'.

R : C ↦ C'

In the diagrams below R is shown as a π/2 radians rotation about the y-axis:

Coordinate system C

             y-axis
                |       /         Q
                |      /          |
                |     /           |
                |    /            |
                |   /             |
                |  /             /
                | /             /
                |/             /
               _/_ _ _ _ _ _ _/_ _ _ _ _ _
              O/                         x-axis
              /         P
             /          |
            /           |
           /            |
          /- - - - - - -|
         /
        /
      z-axis

Coordinate system C'

             y'-axis     x'-axis
                |         /
                |        /
                |       /         Q'
                |      /          |
                |     /           |
                |    /            |
                |   /             |
                |  /             /
                | /             /
                |/             /
               _/_ _ _ _ _ _ _/_ _ _ _ _ _
              O/                         z'-axis
              /         P'
             /          |
            /           |
           /            |
          /- - - - - - -|
         /
        /

In the diagrams:

O is the origin and does not move.
R maps the x/y/z axes to the x'/y'/z' axes respectively.
R⁻¹ (the inverse of R) transforms Q to P in C and therefore transforms Q' to P' in C'.

Suppose we wish to find the coordinates of P in C', that is we wish to find P'. (This is the change of basis we require.)

We have that


R⁻¹: Q' ↦ P'

Since Q' has the same coordinates in C' as P in C, to find P' we simply need to apply R⁻¹ to P:


Q' == P ∴ R⁻¹: P ↦ P'

Finding R⁻¹

The columns of the matrix R are given by 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 ↦ P'

fospathi/change_of_basis.md

Change of basis

Finding R⁻¹