6.6 Test for Linear Dependence

#Type/Theorem #Topic/Linear_Algebra

Types: N/A
Examples: 6.6.1 Using Test for Linear Dependence
Constructions: N/A
Generalizations: 6.4 Linearly Dependent Vectors

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Test for Linear Dependence

Let $m, n \in N$ . Suppose that ${\vec{a}}_{1}, {\vec{a}}_{2}, \dots, {\vec{a}}_{n} \in R^{m}$ . These vectors are 6.4 Linearly Dependent Vectors iff there exists a set of scalars $x_{1}, x_{2}, \dots, x_{n}$ that are not all zero such that

x_{1} {\vec{a}}_{1} + x_{2} {\vec{a}}_{2} + \dots + x_{n} {\vec{a}}_{n} = \vec{0}

with at least one scalar $x_{i} \neq 0$ .