Linear combinations

We shall say, whenever , that is a linear combination of ; we shall use without any further explanation all the simple grammatical implications of this terminology. Thus we shall say, in case is a linear combination of , that is linearly dependent on ; we shall leave to the reader the proof that if is linearly independent, then a necessary and sufficient condition that be a linear combination of is that the enlarged set, obtained by adjoining to , be linearly dependent. Note that, in accordance with the definition of an empty sum, the origin is a linear combination of the empty set of vectors; it is, moreover, the only vector with this property.

The following theorem is the fundamental result concerning linear dependence.

Theorem 1. The set of non-zero vectors is linearly dependent if and only if some , , is a linear combination of the preceding ones.

Proof. Let us suppose that the vectors are linearly dependent, and let be the first integer between and for which are linearly dependent. (If worse comes to worst, our assumption assures us that will do.) Then for a suitable set of ’s (not all zero); moreover, whatever the ’s, we cannot have , for then we should have a linear dependence relation among , contrary to the definition of . Hence as was to be proved. This proves the necessity of our condition; sufficiency is clear since, as we remarked before, every set containing a linearly dependent set is itself such. ◻

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