Chapter 4 - Vector Spaces - 4.5 Basis and Dimension - 4.5 Exercises - Page 188: 71

(a) False. (b) True.

(a) False. Because if If there exists a set of $n-1$ vectors in $V$ that span $V $, then by Theorem 4.12 this set will be a basis and hence the dimension is $n-1$ which is a contradiction. (b) True. Adding any vector to the basis will span $V$. Hence, we can find $n+1$ vectors that span $V$.