The <normal equation for a linear inverse problem> is
$$
\boxed{A^*Au=A^*f}.
$$
It has a solution exactly when
$$
f\in\operatorname{ran}A\oplus(\operatorname{ran}A)^\perp
=\operatorname{dom}(A^\dagger).
$$
When solutions exist they form
$$
A^\dagger f+\ker A.
$$
They are unique exactly when $\ker A=\{0\}$, while $A^\dagger f$ is always the unique solution in $(\ker A)^\perp$ and the solution of minimum norm. Here $A^\dagger$ is the <Moore-Penrose inverse>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-326/3/c/solution