= Solution
{c}
Suppose $\mathbf x_0$ is one solution. By <linearity>,
$$
A\mathbf x=\mathbf b
\iff A(\mathbf x-\mathbf x_0)=0
\iff\mathbf x-\mathbf x_0\in\ker T_A.
$$
Thus the <solution set of a linear system> is exactly $\mathbf x_0+\ker T_A$. It has one member if the <kernel> is $\{0\}$. Conversely, if there is a unique solution and $\mathbf z\in\ker T_A$, then $\mathbf x_0+\mathbf z$ is also a solution, forcing $\mathbf z=0$. Combining this with <2/a/i> gives
$$
\boxed{\text{unique solution}\iff\mathbf b\in\operatorname{im}T_A\text{ and }\ker T_A=\{0\}.}
$$
Back to article page
