Representing a Vector With Respect to a Change of Basis
A transformation may be represented by a matrixoperating on a vectorwhere is the position vector of a point P. The resulting transformed position vector is
It is important to note that the position of the pointdoes not change in space, only the representation of the point with respect to the new coordinate system. We may consider the matrix acting to change the basis of the space into a new basis.
Suppose the transformation T is represented by the matrixIf the original coordinate system has basis vectorscalled the standard basis
then the transformed basis vectors are given byand
We may write a point with position vectoras
We can find a similar representation in the new coordinate system for a vectorin terms ofand
Suppose we have a vectorWe can writeso
This is general. If the matrix representing a change of basis isthe basis vectorsandtransform asandbut the componentsandtransform asand respectively.