Answer
a. $n$
b. By the equivalence of vector and matrix equations, and by the definition of linear independence*, we know that the columns of a matrix are linearly independent if and only if $\mathbf{A}\vec{x}=\vec{0}$ has only the trivial solution. But, as indicated in the box on page 44, this is the case if and only if the equation has no free variables, i.e., if every column is a pivot column. Hence, since an $m\times n$ matrix has $n$ columns, all $n$ must be pivot columns.
Work Step by Step
* Recall that a set $\{\vec{v}_{1},...,\vec{v}_{n}\}$ of vectors is linearly dependent if and only if $c_{1}\vec{v}_{1}+...+c_{n}\vec{v}_{n}=\vec{0}$ has a solution such that not all $c_{1},...,c_{n}$ are equal to zero, and that a set of vectors is linearly independent if and only if it is not linearly dependent.
