We define the variance ellipse of the random
variable y with 
and 
as
so that the variance ellipse of any 
is
The image of 
under A is
and we show that 
and that 
.
Note first that because 
and 
, for all z=Ay there is a b
such that 
. Therefore, given 
consider 
. This y is a member of 
:
Hence, 
. Also, for all 
, let 
. There is a b such that Ay= 
. Because 
, this b will
satisfy the inequality
Hence, 
and 
. In other words, 
.