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A source is nonradiating at frequency $k$ if its outgoing field has zero far-field pattern. For every $\chi \in C_c^{\infty }(A)$, the source $Q=(\Delta +k^2)\chi$ has outgoing field $\psi =\chi$ and is invisible outside $A$. Integration by parts shows that its Fourier transform vanishes on $|\mathbf{\xi }|=k$, so it belongs to the kernel of a linear map defined by the source-to-far-field operator at fixed frequency. This family proves nonuniqueness independently of the instability of the inverse.

For a matrix whose entries depend on a parameter, a nonzero maximal minor certifies a fixed matrix rank away from its zeros. At a zero, other minors must still be checked: one vanishing minor does not itself imply a rank drop. Once the matrix rank is determined, the rank-nullity theorem gives the dimension of the kernel of a linear map. Exceptional parameters can then be solved separately to exhibit the additional kernel of a linear map directions.

For any field $F$ and vector subspace $W\subseteq F^m$, some $u\in W$ has at least $\dim W$ nonzero coordinates. Choose $u$ with largest support of a vector $S$. Restriction from $W$ to $F^S$ is injective: a nonzero vector in its kernel of a linear map vanishes on $S$, so adding it to $u$ would enlarge $S$. Thus $\dim W\le |S|$. The argument works over finite fields, where assuming a generic vector avoids finitely many hyperplanes would be invalid.