= Noninjective linear measurement
{title2=$T(x)=T(x')\iff x-x'\in\ker T$}
When a measurement model is a <linear map> $T$, its <kernel> records changes that its readout cannot distinguish. For example, the <rectangular kernel and solution-fibre example> measures only $x+2y$ and $y+z$. Even exact readouts $(3,2)$ leave the whole <fiber> $u_0+\ker A$ possible. Repeating either readout with greater precision cannot distinguish those inputs: their outputs agree exactly. This ambiguity follows from a nontrivial <kernel>, rather than from measurement error.
Additional information can remove a <null direction of a linear map>. In that example, adding a third <linear functional> that reads $y$ distinguishes $u_0+tv$, because its third readout is $t$. More generally, additional <linear functionals> $c_1,\ldots,c_k$ give unique recovery of all inputs exactly when
$$
\ker T\cap\ker c_1\cap\cdots\cap\ker c_k=\{0\}.
$$
This follows by applying the <trivial kernel> criterion to the combined <linear map> $x\mapsto(Tx,c_1(x),\ldots,c_k(x))$. Merely adding repeated <linear equations> does not meet this condition. On a finite-dimensional input <vector space>, each added <scalar> readout can increase the <matrix rank> by at most one, so at least $\dim\ker T$ extra independent readouts are needed for recovery of unrestricted inputs.
Restrictions on the possible inputs can also make recovery unique. If the allowed inputs form a <set> $D$, two allowed inputs are indistinguishable exactly when their difference is in the <kernel>. Thus $T$ is <injective> on $D$ precisely when no difference of distinct points of $D$ is a <kernel> <vector>. Such a restriction is extra information about the input; the original readout alone determines only its <fiber>.
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