Chapter 1.5 Definitions (Linear Dependence and Linear Independence)

All definitions in Chapter 1.5

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 + a2u2 + … + anun = 0

We also say that the vectors of S are ______ ________.

Trivial Representation (of 0)

If for any vectors u1, u2, …, un, we have

a1u1 + a2u2 + … + anun = 0

if a1=a2=…=an=0

For a set to be linearly dependent, there must exist a nontrivial representation of 0 as a linear combination of the vectors in the set.

Linearly Independent

A subset S of a vector space V that is not linearly dependent is called ______ ________.

We also say that the vectors of S are ______ ________.

Facts About Linearly Independent Sets in any Vector Space

1) The empty set is always linearly independent since linearly dependent sets must be nonempty

2) A set consisting of a single nonzero vector is linearly independent

3) A set is linearly independent if and only if the only representations of 0 as linear combinations of its vectors are trivial representations (so basically when it’s a Trivial Representation of 0)

Theorem 1.6

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

Theorem 1.7

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