6.6.1 Using Test for Linear Dependence

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Generalizations: 6.6 Test for Linear Dependence

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Determine if the set of vectors is linearly dependent using the 6.6 Test for Linear Dependence theorem.

\begin{aligned} {\vec{a}}_{1} = [\begin{array}{c} 5 \\ 0 \\ 0 \\ 5 \end{array}] & {\vec{a}}_{2} = [\begin{array}{c} 0 \\ - 1 \\ - 1 \\ 0 \end{array}] & {\vec{a}}_{3} = [\begin{array}{c} 4 \\ 4 \\ 4 \\ 4 \end{array}] \end{aligned}

Solution. We must find some set of scalars that would create a linear combation from this set of vectors equal to $\vec{0}$ . For this example, we will use the set of scalars $- \frac{1}{5}, 1, \frac{1}{4}$ .

\begin{aligned} - \frac{1}{5} [\begin{array}{c} 5 \\ 0 \\ 0 \\ 5 \end{array}] + 1 [\begin{array}{c} 0 \\ - 1 \\ - 1 \\ 0 \end{array}] + \frac{1}{4} [\begin{array}{c} 4 \\ 4 \\ 4 \\ 4 \end{array}] & = [\begin{array}{c} - 1 \\ 0 \\ 0 \\ - 1 \end{array}] + [\begin{array}{c} 0 \\ - 1 \\ - 1 \\ 0 \end{array}] + [\begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \end{array}] \\ = [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}] \end{aligned}

Therefore, the set of vectors is indeed linearly dependent.