Question

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

The set is linearly dependent.

Step-by-step solution

Determine whether the vectors are multiples of each other

Observe that the vectors are not multiples of each other.

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.

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.