A set is linearly independent iff all its finite subsets are also L.I.

up:: 021a MOC Linear Algebra

Let be a Vector Space, and . Then we have that is Linearly Independent iff all of its subsets are linearly independent.

This follows from the contrapositive of A set is linearly dependent iff there is a vector which is a linear combination of previous vectors (provided adequate rearrangements of the set ).

Constructive proof

Entire set is L.I. All finite subsets are L.I.

Let be linearly independent. Then, for all finite combinations of , we’ll have that their Linear Combination will only be zero when all of them are multiplied by ─ which makes all of these finite subsets L.I. in their own right.

All finite subsets are L.I. Entire set is L.I.

Proof by contrapositive: assume the entire set is Linearly Dependent. Then there is some finite combination of vectors in such that, for at least some , , for which we have

Thus, there is some subset which is L.D..