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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

In Exercises 31, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

31. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\4&{ - 1}&3&{ - 6}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\0&7&{ - 1}&{ - 6}\end{array}} \right]\)

Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

29. \(\left[ {\begin{array}{*{20}{c}}0&{ - 2}&5\\1&4&{ - 7}\\3&{ - 1}&6\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&4&{ - 7}\\0&{ - 2}&5\\3&{ - 1}&6\end{array}} \right]\)

Use the accompanying figure to write each vector listed in Exercises 7 and 8 as a linear combination of u and v. Is every vector in \({\mathbb{R}^2}\) a linear combination of u and v?

8.Vectors w, x, y, and z

Give a geometric description of Span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors in Exercise 16.

Explain why a set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},{{\mathop{\rm v}\nolimits} _4}} \right\}\) in \({\mathbb{R}^5}\) must be linearly independent when \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\) is linearly independent and \({{\mathop{\rm v}\nolimits} _4}\) is not in Span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).
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