= Solution
The homogeneous <linear system> has <solution set> $\ker A$, a <vector subspace>. If it contains only the zero vector, no two different solutions of the full system can exist: their difference would be a nonzero homogeneous solution.
A <particular solution> $\mathbf v_p$ to the full system exists exactly when $\mathbf w$ belongs to the <image of a linear map> $\operatorname{im}A$. If it exists, every solution is
$$
\boxed{\mathbf v=\mathbf v_p+\mathbf v_h,\qquad\mathbf v_h\in\ker A}.
$$
Indeed, subtracting $A\mathbf v_p=\mathbf w$ from $A\mathbf v=\mathbf w$ gives $A(\mathbf v-\mathbf v_p)=0$, and conversely adding a <kernel> vector preserves the right-hand side. Thus there are \b[no solutions], \b[one solution] if the <kernel> is trivial, or \b[infinitely many solutions] if it is nontrivial. The last <solution set> is an <affine subspace> of <dimension> $\dim\ker A$, determined by the <rank-nullity theorem>. For a square nonsingular matrix there is one solution for every right-hand side.
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