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  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ be an $m \\times n$ matrix over $F$.\nThe *null space* of $A$ is\n$$\\operatorname{Null}(A) = \\{x \\in F^n : Ax = 0\\}.$$\nIt is the [[kernel|0S23E6YE83]] of the linear map $x \\mapsto Ax$.\n\n## Remarks\n\nThe null space is a subspace of $F^n$, of dimension $n - \\rank A$, the [[nullity|0APCFW7639]] of $A$.\nRow equivalent matrices have the same null space.\nThe null space of $\\begin{pmatrix} 1 \u0026 1 \\end{pmatrix}$ is the line spanned by $(1, -1)^T$.\nEach vector of the null space is orthogonal to each row of $A$, for the dot product.\n\n## Explanations\n\n### L1 human\n\nThe null space of a matrix is the set of solutions of $Ax = 0$.\nWe find a basis of it by row reduction: each free variable gives one basis vector.\nIt is the kernel of the matrix, viewed as a linear map.\n",
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