6.4.2 Examples of Linear Dependence

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Generalizations: 6.4 Linearly Dependent Vectors

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Linear Dependence of Vectors

Consider the vertex vectors

\begin{aligned} {\vec{v}}_{1} = [\begin{array}{c} 0 \\ 0 \end{array}], & {\vec{v}}_{2} = [\begin{array}{c} - 1 \\ 1 \end{array}], & {\vec{v}}_{3} [\begin{array}{c} 2 \\ 1 \end{array}] . \end{aligned}

This set of vectors is linearly dependent because

{\vec{v}}_{1} = 0 \cdot {\vec{v}}_{2} + 0 \cdot {\vec{v}}_{3} .

In other words, we can write ${\vec{v}}_{1}$ as a linear combination of ${\vec{v}}_{2}$ and ${\vec{v}}_{3}$ . Additionally, any set of vectors that includes the zero vector will always be linearly dependent.

Linear Dependence of Vectors

Consider the set of vectors.

\begin{aligned} {\vec{a}}_{1} = [\begin{array}{c} 1 \\ - 3 \end{array}], & {\vec{a}}_{2} = [\begin{array}{c} - 3 \\ 9 \end{array}], & {\vec{a}}_{3} = [\begin{array}{c} 4 \\ 9 \end{array}] \end{aligned}

The set of vectors is linearly dependent. We can see this immediately from the fact that

{\vec{a}}_{1} = - 3 \cdot {\vec{a}}_{2} .