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Chapter 1: Q16E (page 1)

Determine by inspection whether the vectors in Exercises 15-20 are linearly independent. Justify each answer.

16. \(\left[ {\begin{array}{*{20}{c}}4\\{ - 2}\\6\end{array}} \right],\left[ {\begin{array}{*{20}{c}}6\\{ - 3}\\9\end{array}} \right]\)

Short Answer

Expert verified

The set is linearly dependent.

Step by step solution

01

Determine whether the vectors are multiples of each other

Write the two \({{\mathop{\rm v}\nolimits} _2}\) vectors in the expression \({{\mathop{\rm v}\nolimits} _1}\) as shown below:

\(\begin{aligned}{c}{v_2} &= \left[ {\begin{array}{*{20}{c}}6\\{ - 3}\\9\end{array}} \right]\\ &= \frac{3}{2}\left[ {\begin{array}{*{20}{c}}4\\{ - 2}\\6\end{array}} \right]\\ &= \frac{3}{2}{{\mathop{\rm v}\nolimits} _1}\end{aligned}\)

02

Determine whether the vectors are linearly independent

A set of two vectors \(\left\{ {{v_1},{v_2}} \right\}\)islinearly dependentif at least one of the vectors is a multiple of the other. The set islinearly independent if and only if neither of the vectors is a multiple of the other.

Here, the second vector is \(\frac{3}{2}\) times the first vector.

Thus, the set is linearly dependent.

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Most popular questions from this chapter


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