6.8 Test for Linear Independence

#Type/Theorem #Topic/Linear_Algebra

Types: N/A
Examples: 6.7.1 Fantasy Basis (Elementary Standard Basis for R5
Constructions: N/A
Generalizations: 6.7 Linearly Independent Vectors

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

Test for Linear Independence

Let $m, n \in N$ . Suppose vectors ${\vec{a}}_{1} {\vec{a}}_{2}, \dots, {\vec{a}}_{n} \in R^{m}$ . These vectors are 6.7 Linearly Independent Vectors iff

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

is true ONLY when $x_{k} = 0$ for all $k = 1, 2, \dots, n$ .