Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 1 / 6A / ii / Solution

If $Ax=b$ and $h\in \ker A$, symmetry gives $b\cdot h=(Ax)\cdot h=x\cdot Ah=0$, proving necessity. Conversely, expand $b$ in the orthonormal basis from part (i), with $A{y}_i={\lambda }_i{y}_i$. The zero-eigenvalue vectors span the kernel of a linear map. If $b$ is orthogonal to that kernel, then $b\cdot y_i=0$ whenever ${\lambda }_i=0$, so the remaining coordinates can be solved independently.

For a nonzero eigenvalue and any associated nonzero eigenvector $v$, symmetry gives $b\cdot v=\lambda x\cdot v$. Thus the component vector along $v$ isFor the unit vectors $y_i$, this coefficient is $(b\cdot y_i)/{\lambda }_i$. Consequently the general solution isThis proves sufficiency as well as the claimed kernel criterion. If the compatibility condition fails, there is no solution; if it holds, the freedom is exactly an arbitrary kernel vector.