Chapter 1.5 Definitions (Linear Dependence and Independence)

Linearly Dependent

A subset S of a vector space V is called _____ _______ if there exist a finite number of distinct vectors u1, u2, …, un in S and scalars a1, a2, …, an not all zero such that

a1u1 + … + anun = 0

Trivial Representation

For any vectors u1, …, un we have a1u1 + … + anun = 0 if a1=a2=…=an=0.

Linearly Independent

If a subset S of a vector space is not linearly dependent; in other words, can only be written as the trivial representation

Theorem 1.6

Let V be a vector space, and let S1⊆S2⊆V. If S1 is linearly independent, then S2 is linearly independent.

Corollary: If S2 is linearly independent, then S1 is linearly independent.

Theorem 1.7

Let S be a linearly independent subset of a vector space V, and let v be a vector in V that is not in S. Then S∪{v} is linearly dependent iff v∈span(S)