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  "body": "\n## Statement\n\nLet $A$ be an $m \\times n$ matrix with columns $v_1, \\dots, v_n$. Let $x = (x_1, \\dots, x_n)^T$ be a column vector with $n$ entries, and let $b$ be a column vector with $m$ entries.\n\nThen the following three statements are equivalent.\n\n1. The vector $x$ satisfies the equation $Ax = b$, using the [[matrix-vector product|0PDVQEP0ZC]].\n2. The vector $b$ is the [[linear combination|0V8F3E48E6]] $x_1 v_1 + \\dots + x_n v_n$ of the columns of $A$, with the entries of $x$ as weights.\n3. The entries $x_1, \\dots, x_n$ form a solution of the [[linear system|0GT4WWGWH2]] whose [[augmented matrix|0T6MWGZSXA]] is $[v_1 \\; \\dots \\; v_n \\mid b]$.\n\n## Remarks\n\nThe product $Ax$ is defined as the combination of the columns of $A$ weighted by the entries of $x$. This gives the link between the first two statements. The third statement reads the same equation one row at a time.\n",
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