Suppose $A$ is a $n \times (n+1)$ matrix of rank $n$.

(a) Show that the one dimensional solution space of $A x =b$ varies continuously with $b \in \mathbb{R}^n$.

(b) Generalize

My attempt:

(a) The nullspace of $A$ has dimension 1. If I think of the RREF of $A$ obtained by premultiplication with an $n \times n$ matrix $E$, then the solution to $Ax= b$ is

$x = \begin{pmatrix} Eb \\ 0 \\ \end{pmatrix} + \alpha \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ where the second term on the right is a general vector in the nullspace of $A$.
Thus for RHS $b_1, b_2$, (now denoting the nullspace basis vector by $e_n$

$||x_1 - x_2|| \leq ||E|| \cdot||b_1-b_2|| + |\gamma| \cdot ||e_n||, \forall \; \gamma \;\in \mathbb{R} $.

This would prove the claim if $\gamma=0$, but as it stands, I do not see if this is correct. How do I show the claim ?

(b) I can think of

  1. Rank of A = n, but dimension of nullspace > 1.

  2. Rank of A < n. In this case it is not clear to me how to even begin, since there may be no solution for a given $b \in \mathbb{R}^n$

So I am puzzled by the point of this question.


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