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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \\colon V \\to W$ be a [[linear map|0NRYHJPQG2]].\nThe *kernel* of $T$ is\n$$\\ker T = \\{v \\in V : T(v) = 0\\}.$$\n\n## Remarks\n\nThe kernel is a [[linear subspace|00YC1D652K]] of $V$.\n[[A linear map is injective if and only if its kernel is trivial|0BJBYCGMWE]].\nFor the map $x \\mapsto Ax$ of a matrix $A$, the kernel is the [[null space|0PHDYNR24D]] of $A$, the solution set of $Ax = 0$.\nThe kernel of the projection $\\R^3 \\to \\R^2$, $(x, y, z) \\mapsto (x, y)$, is the $z$-axis.\n\n## Explanations\n\n### L1 human\n\nThe kernel collects the vectors that the map sends to zero.\nIt measures how much information the map loses.\nIf only the zero vector goes to zero, then the map loses nothing.\n",
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