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  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ be an $m \\times n$ matrix over $F$ with columns $a_1, \\dots, a_n \\in F^m$.\nThe *column space* of $A$ is\n$$\\operatorname{Col}(A) = \\operatorname{span}(a_1, \\dots, a_n) \\subseteq F^m,$$\nthe [[linear span|0DTY8B9HMS]] of the columns.\n\n## Remarks\n\nThe column space is $\\{Ax : x \\in F^n\\}$, the image of the map $x \\mapsto Ax$.\nThe system $Ax = b$ is [[consistent|0X6V9B1V10]] if and only if $b \\in \\operatorname{Col}(A)$.\nThe dimension of the column space is the [[rank|0ZC9YTXZRY]] of $A$.\nThe pivot columns of $A$ form a basis of $\\operatorname{Col}(A)$.\n\n## Explanations\n\n### L1 human\n\nThe column space collects all the combinations of the columns of a matrix.\nThese are exactly the vectors $b$ for which $Ax = b$ has a solution.\nFor the matrix $\\begin{pmatrix} 1 \u0026 2 \\\\ 2 \u0026 4 \\end{pmatrix}$, the column space is the line spanned by $(1, 2)$.\n",
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