We consider with the standard basis , its
dual basis
, and the basis consisting in
and
.
We want to express the dual basis
and
as a linear combination of the standard dual basis, that is, we want to determine the coefficients
and
(and and )
in
-
(and in ).
Here,
and .
In order to compute this, we have to express
and
as a linear combination of
and .
This is
-
and
-
Therefore, we have
-
and
-
Hence,
-
With similar computations we get
-
The
transformation matrix
from to is thus
-
The transposed matrix of this is
-
The inverse task to express the standard dual basis with
and ,
is easier to solve, because we can read of directly the representations of the with respect to the standard basis. We have
-
and
-
as becomes clear by evaluation on both sides.