Let's have the following linear system $Ax = b$ and assume that $x_0$ is a solution this linear system, where $A$ is an $n \times m$ matrix, with $n < m$, $x$ is an $m$-dimensional column vector and $b$ is an $n$-dimensional column vector.
Let's have a vector $x_1$ which is orthogonal to the solution $x_0$. Is $x_1$ in a null space of $A$ ?
I want to prove that $x_0 + x_1$ is also a solution to $Ax = b$ and for this, $x_1$ has to be in the null space of $A$, so that this holds:
$$A(x_0 + x_1 )= b $$ $$Ax_0 + Ax_1 = b $$
$$b + Ax_1 = b $$ $$ Ax_1 = 0 $$
How to find whether the last equation holds?