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Chapter 1: Q15E (page 1)

Determine by inspection whether the vectors are linearly dependent. Justify each answer.

\(\left[ {\begin{array}{*{20}{c}}5\\1\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\8\end{array}} \right],\left[ {\begin{array}{*{20}{c}}1\\3\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{ - 1}\\7\end{array}} \right]\)

Short Answer

Expert verified

The vectors are linearly dependent.

Step by step solution

01

Set of two or more vectors

When a set has more vectors than entries in each vector, it is said to be linearly dependent.

Let \({v_1},{v_2},\,{v_3}\), and \({v_4}\) be the four vectors. The linear dependence of these three vectors in the form of an augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{v_1}}&{{v_2}}&{{v_3}}&{{v_4}}&0\end{array}} \right]\).

02

Augmented matrix

The augmented matrix is represented as:

\(\left[ {\begin{array}{*{20}{c}}5&2&1&{ - 1}&0\\1&8&3&7&0\end{array}} \right]\)

03

Linearly independent vector

There are four vectors given, but there are only two entries in each of the vectors. We know that if a set has more vectors than entries in each vector, it is linearly dependent.

Hence, the vectors are linearly dependent.

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Most popular questions from this chapter

In Exercises 15 and 16, list five vectors in Span \(\left\{ {{v_1},{v_2}} \right\}\). For each vector, show the weights on \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) used to generate the vector and list the three entries of the vector. Do not make a sketch.

16. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\)

Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot position in each column. Explain why the system has a unique solution.

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

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

Construct three different augmented matrices for linear systems whose solution set is \({x_1} = - 2,{x_2} = 1,{x_3} = 0\).

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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