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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 *rank* of $A$ is\n$$\\rank A = \\dim \\operatorname{Col}(A),$$\nthe [[dimension|03GSZ8RTZJ]] of the [[column space|03AJKXH1AV]] of $A$.\n\n## Remarks\n\nThe rank is the maximal number of linearly independent columns of $A$.\nIt also equals the dimension of the row space, by [[row rank equals column rank|0TZAG60ZYT]].\nIt equals the number of pivots in a row echelon form of $A$, and the largest order of a nonzero minor of $A$.\nAn $n \\times n$ matrix is invertible if and only if its rank is $n$.\n\n## Explanations\n\n### L1 human\n\nThe rank of a matrix counts how many of its columns are really independent.\nFor $\\begin{pmatrix} 1 \u0026 2 \\\\ 2 \u0026 4 \\end{pmatrix}$, the second column is twice the first, so the rank is $1$.\nWe compute the rank by row reduction and count the pivots.\n",
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