Find all solutions of least squares problem

Question

I have the following exercise (this is exercise 4.39 of Fundamentals of Matrix Comuptations - Watkins) :

I am not sure about how to find all the solutions(item e). I think I must use itens c) and d) but I don't see how to do it . I found the minimal norm solution :

$x_{mn} = A^{\dagger}$ b = $\big(\begin{smallmatrix} 3/35\\ 6/35 \end{smallmatrix}\big)$

The last column of $V$ is an orthonormal basis for $ \mathcal{N}(A)$ and this column is $\big(\begin{smallmatrix} -2/\sqrt5\\ 1/\sqrt5 \end{smallmatrix}\big)$

Any help will be aprecciated

Answer 1

In this answer I explained that the set of all solutions of the least-squares problem can be written as $\{x_{\text{mn}} + z : z \in \mathcal{N}(A) \}$. Here, $x_{\text{mn}} = A^+ b$ is the minimum-norm solution and $\mathcal{N}(A)$ is the kernel of $A$.

You were able to calculate $x_{\text{mn}} = \frac{1}{35}(3, 6)$ and you obtained that the vector $\frac{1}{\sqrt{5}} (-2, 1)$ spans the kernel of $A$. Therefore, the desired solution set is

$$\{x_{\text{mn}} + z : z \in \mathcal{N}(A) \} = \{(3/35, 6/35) + t ( -2, 1) : t \in \mathbb{R} \}.$$