Suppose **M** is the matrix that describes T in a given basis B, so that
the columns of **M** represent the images of the members of B expressed as
linear combinations of the members of B, and **M** is the matrix similarly
describing T with respect to basis B.

What is the relation of **M** to **M**?

Let** J** be the (Jacobian) matrix whose columns are the basis vectors of
B expressed in terms of those of B.

Then **MJ** has columns which are the images of the basis vectors of B expressed
in terms of those of B.

To reexpress these in terms of the basis vectors of B you must multiply on the
left by the matrix which expresses the members of B as linear combinations of
those of B.

This is the inverse Jacobian, **J**^{-1}. We therefore have **M
**=** J**^{-1} MJ, and, by our last result, as claimed:

**M**=**J**^{-1}**M****J**
=** ****J**^{-1}J**M****=****I****M****=****M**.