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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

Hence

This is general. If the matrix representing a change of basis isthe basis vectorsandtransform asandbut the componentsandtransform asand respectively.