In Exercises 29 and 30, V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. Mark each statement True or False. Justify each answer. (These questions are some difficult than those 19 and 20.)

30.

a. If there exists a linearly dependent set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) in \(V\), then \(\dim V \le p\).

b. If every set of \(p\) elements in \(V\) fails to span \(V\), then \(\dim V > p\).

c. If \(p \ge 2\), and \(\dim V = p\), then every set of \(p - 1\) nonzero vectors is linearly independent.

a)

Consider the set \(\left\{ {{\mathop{\rm v}\nolimits} ,2{\mathop{\rm v}\nolimits} } \right\}\) with \({\mathop{\rm v}\nolimits} \) as a non-zero vector in \({\mathbb{R}^3}\). The set is linearly dependent in \({\mathbb{R}^3}\), but \(\dim {\mathbb{R}^3} = 3 > 2\).

Thus, statement (a) is false.

b)

The basis for \(V\) has \({\mathop{\rm p}\nolimits} \) vectors or fewer vectors. The basis spans the set for \(V\) with \({\mathop{\rm p}\nolimits} \) vectors or fewer vectors, which contradicts the assumption.

Thus, statement (b) is true.

c)

Consider \({\mathop{\rm v}\nolimits} \) is a non-zero vector in \({\mathbb{R}^3}\) and the set is \[\left\{ {{\mathop{\rm v}\nolimits} ,2{\mathop{\rm v}\nolimits} } \right\}\]. The set is linearly dependent in \({\mathbb{R}^3}\) with \(3 - 1 = 2\) vectors, and \(\dim {\mathbb{R}^3} = 3\).

Thus, statement (c) is false.