Chapter 5, Problem 4CTB

Let A be a $7\times 5$ matrix with rank equal to 4 and let b be a vector in ${ℝ}^8.$ The four fundamental sub-spaces associated with A are $R\left(A\right),N\left(A^T\right),R\left(A^T\right),$ and $N\left(A\right).$
(a)WhatisthedimensionofN(AT),andwhichof the other fundamental subspaces isthe orthogonalcomplementofN(AT)?
(b) IfxisavectorinR(A)and $A^{T}x=0,$ then what can you conclude about the value of $\Vert x\Vert ?$ Explain.
(c) What is the dimension of $N\left(A^{T}A\right)?$ How many solutions will the least squares system $Ax=b$ have? Explain.