ABSTRACT
Definition 5.2 Grey linear combinations: Assume a grey vector group a a⊗ ⊗, ,1 a a a2
⊗ ⊗ ⊗, , , , i min which a a a a ai i i ij in ⊗ ⊗ ⊗ ⊗ ⊗= { , , , , , },1 2 a a aij ij ijL R
⊗ ⊗ ⊗= [ , ], i = 1 2, , , m j n, , , ,= 1 2 are all n-dimensional grey vectors and k k ki1 2
⊗ ⊗ ⊗, , , , , km
⊗ is a group of grey numbers. If every grey component g g ga ai ki ij i m, , , , , , ,= 1 2 j n= 1 2, , , of a a⊗ ⊗ ⊗ =, , , , ,i ik i m1 2 meets the conditions, then
a a a ag g g⊗
We can say that grey vector a⊗ can be grey linear represented by the grey vector group a a a a1 2
⊗ ⊗ ⊗ ⊗, , , , , , i m or a ⊗ is a grey linear combination of a1
⊗ , a a a2
⊗ ⊗ ⊗, , , , . i m
Theorem 5.1 Necessary conditions of g g ga ai ki ij i m j n, , , , , , , , , , := =1 2 1 2 For an arbitrary grey vector a⊗ and an arbitrary grey vector group a1
⊗ , a a a2
⊗ ⊗ ⊗, , , , i m in which a a a a ai i i ij in ⊗ ⊗ ⊗ ⊗ ⊗= { , , , , , },1 2 if a
⊗ can be grey linear represented by a grey vector group a a a a1 2
⊗ ⊗ ⊗ ⊗, , , , , i m that has a grey coefficient group of k k k ki m1 2
⊗ ⊗ ⊗ ⊗, , , , , , we can meet the conditions of g g ga ai ki ij i m j n, , , , , , , , , , ,= =1 2 1 2 which is the obtained number of the unit grey numbers of a a⊗ ⊗ ⊗ =, , , , , .i ik i m1 2
Proof: To make the proof general, for an arbitrary n-dimensional grey vector a⊗ , a grey vector group a a a a1 2
⊗ ⊗ ⊗ ⊗, , , , , , i m and a grey coefficient group k k k ki m1 2
⊗ ⊗ ⊗ ⊗, , , , , , we assume that the obtained number of the unit grey numbers of this grey coefficient group meets the conditions that make Eq. (5.1) tenable.
