Page 370 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

5.3 In the preceding section we learned that a set of vectors

LINEAR INDEPENDENCE spans a given vector space V if every vector in V is expressible as a linear
combination of the vectors in S. In general, there may be more than one way to
express a vector in V as a linear combination of vectors in a spanning set. In
this section we shall study conditions under which each vector in V is
expressible as a linear combination of the spanning vectors in exactly one way.
Spanning sets with this property play a fundamental role in the study of vector
spaces.

DEFINITION

If is a nonempty set of vectors, then the vector equation

has at least one solution, namely

If this is the only solution, then S is called a linearly independent set. If there are other solutions, then S is called a linearly
dependent set.

EXAMPLE 1 A Linearly Dependent Set

If , , and , then the set of vectors is linearly dependent,
since .

EXAMPLE 2 A Linearly Dependent Set .
The polynomials
form a linearly dependent set in since

EXAMPLE 3 Linearly Independent Sets

Consider the vectors , , and in . In terms of components, the vector equation

becomes