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  "body": "\n## Summary\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, and let $v_1, \\dots, v_k \\in V$ be [[linearly independent|0CRMGPAH83]].\nThe *Gram–Schmidt process* defines vectors $u_1, \\dots, u_k$ by $u_1 = v_1$ and, for $2 \\le j \\le k$,\n$$u_j = v_j - \\sum_{i=1}^{j-1} \\frac{\\langle v_j, u_i \\rangle}{\\langle u_i, u_i \\rangle} u_i.$$\nThen the vectors $e_j = u_j / \\norm{u_j}$ form an [[orthonormal family|0RW9TNNT87]] with the same [[span|0DTY8B9HMS]] as $v_1, \\dots, v_j$ for each $j$.\n\n## Remarks\n\nEach step subtracts from $v_j$ its projections on the earlier vectors $u_i$.\nFor $v_1 = (1, 1)$ and $v_2 = (1, 0)$ in $\\R^2$, it gives $u_1 = (1, 1)$ and $u_2 = (1/2, -1/2)$.\nIn floating point arithmetic, the modified Gram–Schmidt process is more stable.\nIn matrix form, the process gives the [[QR decomposition|0ERWG4NXQ9]].\n\n## Explanations\n\n### L1 human\n\nThe Gram–Schmidt process turns any independent list of vectors into a list of perpendicular unit vectors.\nIt treats the vectors one at a time, and it removes from each one its components along the earlier ones.\nThen it scales each result to length one.\n",
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