6.4 Linearly Dependent Vectors

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 6.4.1 Span of 3 Vectors Example, 6.4.2 Examples of Linear Dependence
Constructions: 6.6 Test for Linear Dependence
Generalizations: N/A

Properties: 6.5 Theorem of Vector List Size m x 1
Sufficiencies: N/A
Questions: N/A

Linearly Dependent Vectors

Let $m, n \in N$ . Suppose ${\vec{a}}_{1}, {\vec{a}}_{2}, \dots, {\vec{a}}_{n} \in R^{m}$ . We say that the set of vectors ${{\vec{a}}_{k}}_{k = 1}^{n}$ is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors

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

for some set of scalars ${x_{k}}_{k = 1}^{n} \subseteq R$ .

Remark. Generally, when one vector of a set of vectors is linearly dependent, the whole set is linearly dependent.
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