We consider in the
standard basis,
-
and the basis
-
The basis vectors of can be expressed directly with the standard basis, namely
-
Therefore, we get immediately
-
For example, the vector which has with respect to the
coordinates
, has the coordinates
-
with respect to the standard basis . The transformation matrix is more difficult to compute: We have to write the standard vectors as
linear combinations
of
and .
A direct computation
(solving two linear systems)
yields
-
and
-
Hence,
-