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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$ with two [[bases|0KCQ9Y3QKA]] $B = (b_1, \\dots, b_n)$ and $B' = (b'_1, \\dots, b'_n)$.\nThe *change of basis matrix* from $B$ to $B'$ is the $n \\times n$ matrix $P = (p_{ij})$ defined by\n$$b'_j = \\sum_{i=1}^{n} p_{ij} b_i \\qquad (1 \\le j \\le n).$$\nIts column $j$ is the [[coordinate vector|04CYQHN6MC]] $[b'_j]_B$.\n\n## Remarks\n\nFor each $v \\in V$, $[v]_B = P [v]_{B'}$.\nThe matrix $P$ is invertible, and $P^{-1}$ is the change of basis matrix from $B'$ to $B$.\n$P$ is the matrix of the identity map of $V$ in the bases $B'$ (domain) and $B$ (codomain).\nConventions differ between authors; some call $P^{-1}$ the change of basis matrix.\n\n## Explanations\n\n### L1 human\n\nThe change of basis matrix lists the new basis vectors in the old coordinates.\nIt translates coordinates in the new basis into coordinates in the old basis.\nIts inverse translates in the other direction.\n",
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