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  "body": "\n## Description\n\nLet $F$ be a field and let $n \\ge 1$ be an integer.\nFor $i = 1, \\dots, n$, let $e_i \\in F^n$ be the vector with entry $1$ in position $i$ and $0$ in the other positions.\nThe *standard basis* of the coordinate space $F^n$ is the family $(e_1, \\dots, e_n)$.\nIt is a [[basis|0KCQ9Y3QKA]] of $F^n$, and each $x = (x_1, \\dots, x_n)$ is equal to $x_1 e_1 + \\cdots + x_n e_n$.\n\n## Remarks\n\nThe coordinates of a vector in the standard basis are its entries.\nIn $\\R^3$, the standard basis vectors are often written $\\mathbf{i}, \\mathbf{j}, \\mathbf{k}$.\nThe column $j$ of an $m \\times n$ matrix $A$ is $A e_j$.\nThe standard basis of $\\R^n$ is an orthonormal basis for the dot product.\n\n## Explanations\n\n### L1 human\n\nThe standard basis of $F^n$ consists of the vectors with one entry $1$ and all other entries $0$.\nIn the plane, these are the unit arrows along the $x$-axis and the $y$-axis.\nThe coordinates of a vector in this basis are just its entries.\n",
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        "what": "section Invariants",
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