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Chapter 1: Q18E (page 1)

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

18. \(\left[ {\begin{array}{*{20}{c}}4\\4\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{ - 1}\\3\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\5\end{array}} \right],\left[ {\begin{array}{*{20}{c}}8\\1\end{array}} \right]\)

Short Answer

Expert verified

The set is linearly dependent.

Step by step solution

01

Determine whether the vectors are multiples of each other

The vectors are not multiples of each other.

02

Determine whether the set contains more vectors than the entries

Theorem 8 tells that if a set contains more vectors than entries in each vector, then the set is linearly dependent.

Here, the set is linearly dependent since it contains four vectors, with only two entries in each.

03

Determine whether the vectors are linearly independent

The set contains more vectors than the entries in each vector. Hence, it is linearly dependent.

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Most popular questions from this chapter

Consider a dynamical system x鈫�(t+1)=Ax鈫�(t) with two components. The accompanying sketch shows the initial state vector x鈫�0and two eigen vectors 蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫� of A (with eigen values 位1鈫�and位2鈫� respectively). For the given values of 位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=1,位2鈫�=0.9

Find the general solutions of the systems whose augmented matrices are given in Exercises 10.

10. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 1}&3\\3&{ - 6}&{ - 2}&2\end{array}} \right]\)

Consider a dynamical systemwith two components. The accompanying sketch shows the initial state vectorx鈫�0and two eigen vectors蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫�of A (with eigen values 位1鈫�and位2鈫�respectively). For the given values of位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=0.9,位2鈫�=0.9

Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.

17. \(\left[ {\begin{array}{*{20}{c}}2&3&h\\4&6&7\end{array}} \right]\)

In Exercise 23 and 24, make each statement True or False. Justify each answer.

24.

a. Any list of five real numbers is a vector in \({\mathbb{R}^5}\).

b. The vector \({\mathop{\rm u}\nolimits} \) results when a vector \({\mathop{\rm u}\nolimits} - v\) is added to the vector \({\mathop{\rm v}\nolimits} \).

c. The weights \({{\mathop{\rm c}\nolimits} _1},...,{c_p}\) in a linear combination \({c_1}{v_1} + \cdot \cdot \cdot + {c_p}{v_p}\) cannot all be zero.

d. When are \({\mathop{\rm u}\nolimits} \) nonzero vectors, Span \(\left\{ {u,v} \right\}\) contains the line through \({\mathop{\rm u}\nolimits} \) and the origin.

e. Asking whether the linear system corresponding to an augmented matrix \(\left[ {\begin{array}{*{20}{c}}{{{\rm{a}}_{\rm{1}}}}&{{{\rm{a}}_{\rm{2}}}}&{{{\rm{a}}_{\rm{3}}}}&{\rm{b}}\end{array}} \right]\) has a solution amounts to asking whether \({\mathop{\rm b}\nolimits} \) is in Span\(\left\{ {{a_1},{a_2},{a_3}} \right\}\).
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