We finish this subsection by recognizing that the change of basis matrices
For one direction, if left-multiplication by a matrix changes bases then
the matrix represents an invertible function,
simply because the function is inverted by changing the bases back.
Such a matrix is itself invertible, and so nonsingular.
To finish, we will show that any nonsingular matrix
performs a change of basis operation from any given starting basis
to some ending basis.
Because the matrix is nonsingular, it will Gauss-Jordan reduce to the
identity, so there are elementatry reduction matrices such that
Elementary matrices are invertible and their inverses are also elementary,
so multiplying from the left first
by , then by , etc., gives
as a product of elementary matrices
Thus, we will be done if we show that elementary matrices
change a given basis to another basis, for then
changes to some other basis , and
changes to some , ..., and
the net effect is that changes to .
We will prove this about elementary matrices by covering the three types as
Applying a row-multiplication matrix
changes a representation with respect to
to one with respect to
in this way.
Similarly, left-multiplication by a row-swap matrix
changes a representation with respect to the basis
into one with respect to the basis
in this way.
And, a representation with respect to
changes via left-multiplication by a row-combination matrix
into a representation with respect to
(the definition of reduction matrices specifies that and and so this last one is a basis).
In the next subsection we will
see how to translate among representations of
maps, that is, how to change
The above corollary is a special case of this, where the domain and range are
the same space, and where the map is the identity map.