91Ó°ÊÓ

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.