Exercise 7.5

Let $A$ be an $m\times n$ matrix $(m\ge n)$, and let $A=\widehat{Q}\widehat{R}$ be a reduced QR factorization.
(a) Show that $A$ has rank $n$ if and only if all the diagonal entries of $\widehat{R}$ are nonzero.
(b) Suppose $R$ has $k$ nonzero diagonal entries for some $k$ with $0\le k<n$. What does this imply about the rank of $A$? Exactly $k$? At least $k$? At most $k$? Give a precise answer, and prove it.