operating
on a vector
where
is the position vector of a point P. The resulting transformed
position vector is
University Maths Notes: Matrices and Linear Algebra – 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
point
does
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
matrix
If
the original coordinate system has basis vectors
called
the standard basis
then the transformed basis vectors are given by
and
We may write a point with position vector
as
We can find a similar representation in the new
coordinate system for a vector
in
terms of
and
Suppose we have a vector
We
can write
so
Hence
This is general. If the matrix representing a change of
basis is
the
basis vectors
and
transform
as
and
but
the components
and
transform
as
and
respectively.