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  "body": "\n## Summary\n\nLet $A$ be a real $m \\times n$ matrix and let $b \\in \\R^m$.\nA *least squares solution* of $Ax = b$ is a vector $\\hat x \\in \\R^n$ that minimises $\\norm{Ax - b}$, for the Euclidean norm.\nThe vectors $\\hat x$ that do this are exactly the solutions of the *normal equations*\n$$A^T A \\hat x = A^T b.$$\nIf the columns of $A$ are linearly independent, then $A^T A$ is invertible, and $\\hat x = (A^T A)^{-1} A^T b$ is unique.\n\n## Remarks\n\nThe vector $A \\hat x$ is the [[orthogonal projection|0B8KZRT28M]] of $b$ onto the column space of $A$, see [[the closest point property|0MCDDSCB9Z]].\nThe normal equations say that the residual $b - A\\hat x$ is orthogonal to each column of $A$.\nFitting a line $y = c_0 + c_1 t$ to data points $(t_i, y_i)$ is a least squares problem.\nIn practice, the [[QR decomposition|0ERWG4NXQ9]] gives a more stable method than the normal equations.\n\n## Explanations\n\n### L1 human\n\nA system $Ax = b$ with more equations than unknowns usually has no solution.\nThe least squares method finds the $x$ that makes $Ax$ as close as possible to $b$.\nIt is the standard way to fit a model to noisy data.\n",
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