Suppose
x0 is one solution. By
linearity,
Ax=b⟺A(x−x0)=0⟺x−x0∈kerTA.
Thus the
solution set of a linear system is exactly
x0+kerTA. It has one member if the
kernel is
{0}. Conversely, if there is
a unique solution and
z∈kerTA, then
x0+z is also
a solution,
forcing z=0. Combining this with
i gives
unique solution⟺b∈imTA and kerTA={0}.
