Exercise 4.1.5

(a) If $Ax=b$ has a solution and $A^{T}y=0$, then $y$ is perpendicular to $b.b^{T}y=$ ${(Ax)}^{T}y=x^T\left(A^{T}y\right)=0$. This says again that $C(A)$ is orthogonal to $N\left(A^T\right)$. (b) If $A^{T}y=(1,1,1)$ has a solution, $(1,1,1)$ is a combination of the rows of $A$. It is in the row space and is orthogonal to every $x$ in the nullspace.