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  "body": "\n## Statement\n\nLet $A$ be a matrix. Let $v$ and $w$ be column vectors whose size matches the number of columns of $A$, so that the [[matrix-vector products|0PDVQEP0ZC]] below are defined. Let $c$ be a scalar. Then three facts hold.\n\nFirst, $A0 = 0$, where the zero vector on the left has the size of the columns of $A$ and the zero vector on the right has the size of the rows of $A$.\n\nSecond, $A(cv) = c(Av)$.\n\nThird, $A(v + w) = Av + Aw$.\n\n## Remarks\n\nThe second and third facts say that $v \\mapsto Av$ is a linear map. The first fact follows from the second with $c = 0$.\n",
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  "name": "Linearity of the matrix-vector product",
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