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  "body": "\n## Definition\n\nLet $R$ be a ring, let $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $R$, and let $x = (x_1, \\dots, x_n)^T \\in R^n$ be a column vector.\nThe *matrix–vector product* $Ax$ is the column vector in $R^m$ with entries\n$$(Ax)_i = \\sum_{j=1}^{n} a_{ij} x_j$$\nfor $1 \\le i \\le m$.\n\n## Remarks\n\nIf $A_1, \\dots, A_n$ are the columns of $A$, then $Ax = x_1 A_1 + \\cdots + x_n A_n$, a [[linear combination|0V8F3E48E6]] of the columns.\nA [[system of linear equations|0GT4WWGWH2]] is the equation $Ax = b$.\nThe map $x \\mapsto Ax$ is linear, see [[multiplication by a matrix as a linear map|094V2VBJXJ]].\nFor example, $\\begin{pmatrix} 1 \u0026 2 \\\\ 3 \u0026 4 \\end{pmatrix} \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 3 \\\\ 7 \\end{pmatrix}$.\n\n## Explanations\n\n### L1 human\n\nTo multiply a matrix by a vector, take each row of the matrix in turn.\nMultiply its entries by the entries of the vector, position by position, and add the results.\nEach row gives one entry of the answer.\n",
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