Given $m \times n$ real matrix $A$, where $m
$\mathcal{A}(v):=\{v+w | Aw=0\}=v+W$ where $v$ satisfies $Av=b$.
Does the maximum size of linearly independent set in $\mathcal{A}(v)$ always equal to $\dim W$?
in other words:
Does the maximum numbers of independent solutions of the underdetermined system $Ax=b$ always equal to the nullity of $A$?
I can only seeing this by plotting the solutions.
My idea seems naive:
Suppose that $\dim W=r$, and we have $r+1$-linearly independent vectors in $v+W$ say
$v+x_1,v+x_2,\cdots,v+x_{r+1}$
Can we prove that $x_1,x_2,\cdots,x_{r+1}$ are also linearly independent? (in order to get a contradiction)