Change of basis matrix

Definition

Let $F$ be a field and let $V$ be a vector space over $F$ with two bases $B = (b_1, \dots, b_n)$ and $B' = (b'_1, \dots, b'_n)$. The change of basis matrix from $B$ to $B'$ is the $n \times n$ matrix $P = (p_{ij})$ defined by $$b'_j = \sum_{i=1}^{n} p_{ij} b_i \qquad (1 \le j \le n).$$ Its column $j$ is the coordinate vector $[b'_j]_B$.

Remarks

For each $v \in V$, $[v]_B = P [v]_{B'}$. The matrix $P$ is invertible, and $P^{-1}$ is the change of basis matrix from $B'$ to $B$. $P$ is the matrix of the identity map of $V$ in the bases $B'$ (domain) and $B$ (codomain). Conventions differ between authors; some call $P^{-1}$ the change of basis matrix.

Explanations

L1 human

The change of basis matrix lists the new basis vectors in the old coordinates. It translates coordinates in the new basis into coordinates in the old basis. Its inverse translates in the other direction.