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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

A large apartment building is to be built using modular construction techiniques. The arrangement of apartment on any particular floor is to be chosen from one of three basic floor plans. Plan A has 18 apartments on one floor, including 3 three bedroom units, and 8 one bedroom units. 7 two bedroom units and 8 one bedroom units. Each floor of plan B includes 4 three bedroom units, 4 two bedroom units, and 8 one bedroom units. Each floor of plan C includes 5 three bedroom units, 3 two bedroom units, and 9 one bedroom units. Suppose the building contains a total of \({x_{\bf{1}}}\) floors of plan A, \({x_2}\) floor of plans B, and \({x_{\bf{3}}}\) floors of plan C.

a. What interpretation can be given to the vector \({x_{\bf{1}}}\left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{7}}\\{\bf{8}}\end{aligned}} \right)\)?

b. Write a formal linear combination of vectors that expresses the total numbers of three-, two-, and one-bedroom apartments contained in the building.

c. (M) Is it possible to design the building with exactly 66 three bedroom units, 74 two bedrooms units, and 136 one bedroom units? If so, is there more than one way to do it? Explain your answer.

In Exercises 13 and 14, determine if \(b\) is a linear combination of the vectors formed from the columns of the matrix \(A\).

13. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 4}&2\\0&3&5\\{ - 2}&8&{ - 4}\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}3\\{ - 7}\\{ - 3}\end{array}} \right]\)

In Exercises 5-8, write a matrix equation that determines the loop currents. [M] If MATLAB or another matrix program is available, solve the system for the loop currents.

Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.

6. \(\left( {\begin{aligned}{*{20}{c}}1&{ - 6}&4&0&{ - 1}\\0&2&{ - 7}&0&4\\0&0&1&2&{ - 3}\\0&0&3&1&6\end{aligned}} \right)\)

Mark each statement True or False. Justify each answer.

a. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.

b. The row reduction algorithm applies only to augmented matrices for a linear system.

c. A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.

d. Finding a parametric description of the solution set of a linear system is the same as solving the system.

e. If one row in an echelon form of an augmented matrix is \(\left( {\begin{array}{*{20}{c}}0&0&0&5&0\end{array}} \right)\), then the associated linear system is inconsistent.
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