If we wish, we can define a new base 
(where
and 
are the canonical directions given in the
preceding section) to contain our belief grid, and examine the
adjustment of the corresponding belief structure 
by the belief
structure 
. Amongst other results, we find that the summary of
uncertainty over the belief structure is the same as that for the
belief structure 
:
This is because the elements 
constitute a basis for
the vector space 
, so that the corresponding belief structures
and are equivalent for this purpose. We also obtain
the standardised adjusted expectation formula for each direction,
approximately
A further property here is that just as the original directions
are uncorrelated a priori, so too the adjusted expectations
for the canonical directions, 
are uncorrelated.