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Chapter 1: Q17E (page 1)

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

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

Short Answer

Expert verified

The set is linearly dependent.

Step by step solution

01

Determine whether the vectors are multiples of each other

Observe that 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 linearlydependent.

It does not apply here because the number of vectors does not exceed the number of entries in each vector.

03

Determine whether the vectors are linearly independent by inspection

Theorem 9tells that if a set\(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\)in\({\mathbb{R}^n}\)contains the zero vector, then it is linearly dependent.

Here, the set contains the zero vector. Thus, it is linearly dependent.

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Most popular questions from this chapter

Determine the value(s) of \(a\) such that \(\left\{ {\left( {\begin{aligned}{*{20}{c}}1\\a\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}a\\{a + 2}\end{aligned}} \right)} \right\}\) is linearly independent.

In Exercises 10, write a vector equation that is equivalent tothe given system of equations.

10. \(4{x_1} + {x_2} + 3{x_3} = 9\)

\(\begin{array}{c}{x_1} - 7{x_2} - 2{x_3} = 2\\8{x_1} + 6{x_2} - 5{x_3} = 15\end{array}\)

Suppose Tand U are linear transformations from \({\mathbb{R}^n}\) to \({\mathbb{R}^n}\) such that \(T\left( {U{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\) . Is it true that \(U\left( {T{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\)? Why or why not?

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

14. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 6}\\0&3&7\\1&{ - 2}&5\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}{11}\\{ - 5}\\9\end{array}} \right]\)

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