In Exercises 21 and 22, mark each statement True or False. Justify each answer on the basis of a careful reading of the text.

22.

a. Two vectors are linearly dependent if and only if they lie on a line through the origin.

b. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.

c. If \({\mathop{\rm x}\nolimits} \) and \(y\) are linearly independent, and if \(z\) is in Span \(\left\{ {{\mathop{\rm x}\nolimits} ,y} \right\}\), then \(\left\{ {{\mathop{\rm x}\nolimits} ,y,{\mathop{\rm z}\nolimits} } \right\}\) is linearly dependent.

d. If a set in \({\mathbb{R}^n}\) is linearly dependent, then the set contains more vectors than there are entries in each vector.

a.

In geometric terms, two vectors are linear dependent if and only if they lie on the same line passing through the origin.

Thus, statement (a) is true.

b.

Theorem 8 tells nothing about the case in which the number of vectors in the set does not exceed the number of entries in each vector.

For example, consider the set of two vectors, \(\left[ {\begin{array}{*{20}{c}}4\\{ - 2}\\6\end{array}} \right],\left[ {\begin{array}{*{20}{c}}6\\{ - 3}\\9\end{array}} \right]\). It is linearly dependent because one vector is a scalar multiple of the other.

Thus, statement (b) is false.

c.

Any set \(\left\{ {u,v,w} \right\}\) in \({\mathbb{R}^3}\) with, linearly independent vectors \({\mathop{\rm u}\nolimits} \) and \({\mathop{\rm v}\nolimits} \), is linearly dependent if and only if \({\mathop{\rm w}\nolimits} \)is in the plane spanned by \({\mathop{\rm u}\nolimits} \) and \({\mathop{\rm v}\nolimits} \).

Thus, statement (c) is true.

d.

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

For example, consider the set of two vectors, \(\left[ {\begin{array}{*{20}{c}}3\\1\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}6\\2\end{array}} \right]\). It is linearly dependent because one vector is a scalar multiple of the other.

Thus, statement (d) is false.