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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

In Exercises 9, write a vector equation that is equivalent to

the given system of equations.

9. \({x_2} + 5{x_3} = 0\)

\(\begin{array}{c}4{x_1} + 6{x_2} - {x_3} = 0\\ - {x_1} + 3{x_2} - 8{x_3} = 0\end{array}\)


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Determine whether the statements that follow are true or false, and justify your answer.

15: The systemAx鈬赌=[0001]isinconsistent for all 4脳3 matrices A.

Suppose \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2}} \right\}\) is a linearly independent set in \({\mathbb{R}^n}\). Show that \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2}} \right\}\) is also linearly independent.
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