Exercise 3.5.4

(a) $\left[\begin{array}{cc}1\hfill & 0\hfill \\ 1\hfill & 0\hfill \\ 0\hfill & 1\hfill \end{array}\right]$

(b) Impossible: $r+(n-r)$ must be 3

(c) $\left[\begin{array}{cc}1\hfill & 1\hfill \end{array}\right]$

(d) $\left[\begin{array}{cc}9\hfill & -3\hfill \\ 3\hfill & -1\hfill \end{array}\right]$

(e) Impossible Row space $=$ column space requires $m=n$. Then $m-r=n-$ $r$; nullspaces have the same dimension. Section 4.1 will prove $N(A)$ and $N\left(A^T\right)$ orthogonal to the row and column spaces respectively-here those are the same space.