DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11

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The change-of-basis matrix $P_{\mathcal C\leftarrow\mathcal B}=[\operatorname{id}_V]_{\mathcal B}^{\mathcal C}$

Definition

Let $\mathcal B$ and $\mathcal C$ be ordered bases of the same finite-dimensional vector space $V$ . The change-of-basis matrix from $\mathcal B$ -coordinates to $\mathcal C$ -coordinates is

$P_{\mathcal C\leftarrow\mathcal B}:=[\operatorname{id}_V]_{\mathcal B}^{\mathcal C}.$

Its $j$ -th column is $[b_j]_{\mathcal C}$ . The arrow records the direction in which coordinate columns are converted.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

S. Axler, Linear Algebra Done Right, 4th ed., §3D, result 3.82 (standard reference, not scraped)

The change-of-basis matrix PC←B=[id⁡V]BCP_{\mathcal C\leftarrow\mathcal B}=[\operatorname{id}_V]_{\mathcal B}^{\mathcal C}PC←B​=[idV​]BC​