Change of Basis

Suppose:

  

• We have two coordinate frames for a space, and ,
• Want to change from coordinates relative to to coordinates relative to .

Know

relative to .

Want

the coordinates of P relative to .

• Express each component of in terms of :

  

• The change of coordinates is given by the matrix

  

  Factoring gives

  

• Consider:

  

How

do we get ?

• If is orthonormal:

  

• If is orthogonal:

  

• Otherwise, we have to solve a small system of linear equations, using .
• Change of basis from to Standard Cartesian Frame is trivial (since frame elements normally expressed with respect to Standard Cartesian Frame).

Example:

\

where is the standard coordinate frame.

• Matrices mapping from/to to/from :

  

• Check

  

  

  

Generalization

to 3D is straightforward ...

Example:

• Define two frames:

  

• All coordinates are specified relative to the standard frame in a Cartesian 3 space.
• The standard frame happens to be identical to .
• Note that both and are orthonormal.
• The matrix mapping to is given by

  

• Check this matrix:
  Map each element of ,
  See that we get that element relative to .
  (e.g, )
• Question: What is the matrix mapping from to ?

Notes

 

• On the computer, frame elements usually specified in Standard Frame for space.

  Eg, a frame is given by

  

  relative to Standard Frame.

  Question: What are coordinates of these basis elements relative to F?

• Frames are usually orthonormal.
• A point ``mapped'' by a change of basis does not change;

  We have merely expressed its coordinates relative to a different frame.

Readings: Hearn and Baker, Section 5-5 (not as general as here, though). Red book, 5.9, White book, 5.8