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  "body": "\n## Statement\n\nLet $\\mathcal B = (b_1, \\dots, b_n)$ be an ordered [[basis|0KCQ9Y3QKA]] of $\\R^n$.\nLet $P_{\\mathcal B}$ be the matrix whose columns are $b_1, \\dots, b_n$ in this order.\nThen $P_{\\mathcal B}$ is invertible, and for each vector $x \\in \\R^n$ the [[coordinate vector|0BMVBHTWV2]] $[x]_{\\mathcal B}$ satisfies\n$$x = P_{\\mathcal B}\\,[x]_{\\mathcal B}, \\qquad [x]_{\\mathcal B} = P_{\\mathcal B}^{-1}\\,x.$$\n\n## Remarks\n\nThe product $P_{\\mathcal B}[x]_{\\mathcal B}$ is the combination of the columns of $P_{\\mathcal B}$ with the coordinates as weights. This is the expansion of $x$ in the basis.\nThe second formula follows from the first by multiplying with the inverse matrix.\n",
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