= Kernel of a linear map
{title2=$\ker T=\{v:T(v)=0\}$}
{wiki}
= Nullspace
{synonym}
= Kernel
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= Null space
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The <kernel of a linear map> $T:V\to W$ is the <vector subspace> of inputs sent to zero. <Linearity> makes it a <vector subspace>: if $T(v)=T(w)=0$, then $T(av+bw)=0$ for all <scalars> $a,b$. A <vector> in the <kernel> represents a change that the map cannot detect, since $T(x+v)=T(x)$ for every input $x$.
For $T:\mathbb R^3\to\mathbb R^2$ given by $T(x,y,z)=(x+y,z)$, the zero-output equations are $x+y=0$ and $z=0$. Thus
$$
\ker T=\operatorname{span}\{(1,-1,0)\}.
$$
Increasing the first input while decreasing the second by the same amount leaves the output unchanged. For the <linear functional> $\varphi(x,y)=2x-y$, the <kernel> is the <straight line> $y=2x$ shown in its diagram. A <linear map> is <injective> exactly when its <kernel> is the zero <vector subspace>, because $T(v)=T(w)$ is equivalent to $T(v-w)=0$.
The <kernel> describes invisible changes, rather than all inputs that produce a specified nonzero output. If $T(x_0)=b$, the corresponding <fiber of a function> is the <affine solution space of a linear equation> $x_0+\ker T$. The difference of two inputs in that <fiber> belongs to the <kernel>; an input in that <fiber> need not. In particular, for $b\ne0$ the <fiber> cannot be a <vector subspace>, since it does not contain the <zero vector>.<Interactive Linear Algebra by Dan Margalit and others/Solution Sets>[Margalit and Rabinoff, Interactive Linear Algebra, §2.4.2, Key Observation and Remark: translate the homogeneous solution set by any particular solution]{ref}
For the <linear functional> $S:\mathbb R^2\to\mathbb R$, $S(x,y)=x+2y$, the <kernel> is $\operatorname{span}\{(2,-1)\}$. Each <fiber> $S^{-1}(\{c\})$ is the parallel <straight line> $(c,0)+\operatorname{span}\{(2,-1)\}$. The input <vector space> is two-dimensional, but the output is one <scalar>: a whole <straight line> of inputs collapses to one point of the output <real line>.
\Image[/kernel-rank-one-parallel-fibres.png]
{title=Parallel input fibres of $S(x,y)=x+2y$ map to scalar outputs}
{description=The blue input <straight line> is the <kernel>; the other colored input <straight lines> are its <translations>. The output panel is the one-dimensional <real line>, drawn vertically for alignment. All displayed geometry is exact illustrative mathematics.}
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