The orthogonal projection of Ck onto the e?-space, reads 
Jip dud, 0 1 
u. 3% u; € 
i 
eO. e E ufe e.c 
k jk CU (3.1.1) 
p 
k 
then the projected vector C in the eP-space can be written as 
Similarly the orthogonal projection of the observation vector Ax onto 
the same subspace is 
Lupi Jj i'. qup 19 i^ P Av XD 
e (Ax =u; ee £34 e u: g Bii Ax. -ul Ax = Ay. (3.1.2) 
and the projected vector can be written as 
Tl p 
Ay = 
yo 7 9 Ej. 
Figure 2 depicts these orthogonal projections 
  
  
Fig. 2 The orthogonal projections of Ax and e onto e? -space : 
The projected vectors e may not be linearly independent, it can be 
proved that the following theorem holds /4/. 
The e vectors are linearly independent in the condition space, i.e. eP - 
space, if and only if the vectors Cr? So form a linearly independent set 
in the observation space, i.e. e!-space. 
Let us assume that the Cy are linearly independent, then they can be 
considered as a covariant base of an m—dimensional subspace in the e? - 
space. Hence, the e? -space may be split into two subspaces perpendicular 
to each other, one, spanned by the vectors Cy and the other spanned by 
(b-m) vectors =* perpendicular to the ee (see fig 2). 
The covariant metric tensor of the Cr subspace is 
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