TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$[v]_{\mathcal C}=P_{\mathcal C\leftarrow\mathcal B}[v]_{\mathcal B}$ and $P_{\mathcal B\leftarrow\mathcal C}=P_{\mathcal C\leftarrow\mathcal B}^{-1}$

Statement

For ordered bases $\mathcal B,\mathcal C$ of a finite-dimensional vector space $V$ and every $v\in V$ ,

$[v]_{\mathcal C}=P_{\mathcal C\leftarrow\mathcal B}[v]_{\mathcal B}.$

Moreover,

$P_{\mathcal B\leftarrow\mathcal C}=P_{\mathcal C\leftarrow\mathcal B}^{-1}.$

Facts & Assumptions

Given: Ordered bases $\mathcal B,\mathcal C$ of $V$ and a vector $v\in V$ .

[L1]

$P_{\mathcal C\leftarrow\mathcal B}$ is the matrix of $\operatorname{id}_V$ from $\mathcal B$ -coordinates to $\mathcal C$ -coordinates (The change-of-basis matrix $P_{\mathcal C\leftarrow\mathcal B}=[\operatorname{id}_V]_{\mathcal B}^{\mathcal C}$ ).

[L2]

Coordinate action converts a vector's coordinate column by the matrix of the map, and matrix representation sends composites to products ( $[T(v)]_{\mathcal C}=[T]_{\mathcal B}^{\mathcal C}[v]_{\mathcal B}$ , $[S\circ T]_{\mathcal B}^{\mathcal D}=[S]_{\mathcal C}^{\mathcal D}[T]_{\mathcal B}^{\mathcal C}$ ).

Proof

technique · direct

1.1

Applying coordinate action from [L2] to $\operatorname{id}_V$ and using [L1] gives $[v]_{\mathcal C}=P_{\mathcal C\leftarrow\mathcal B}[v]_{\mathcal B}$ .

givenL1L2

2.1

Represent the identity composition $\operatorname{id}_V\circ\operatorname{id}_V$ first from $\mathcal B$ through $\mathcal C$ back to $\mathcal B$ , and then from $\mathcal C$ through $\mathcal B$ back to $\mathcal C$ .

step 1.1L1L2

3.1

The composite-matrix formula in [L2] gives $P_{\mathcal B\leftarrow\mathcal C}P_{\mathcal C\leftarrow\mathcal B}=I$ and $P_{\mathcal C\leftarrow\mathcal B}P_{\mathcal B\leftarrow\mathcal C}=I$ , so the two matrices are inverses. Empty ordered bases give the same two equations in $M_0(F)$ .

step 2.1L1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 7 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)
S. Schiavone, MIT 18.700 Day 9, Corollary 36 (standard reference, not scraped)

[v]C=PC←B[v]B[v]_{\mathcal C}=P_{\mathcal C\leftarrow\mathcal B}[v]_{\mathcal B}[v]C​=PC←B​[v]B​ and PB←C=PC←B−1P_{\mathcal B\leftarrow\mathcal C}=P_{\mathcal C\leftarrow\mathcal B}^{-1}PB←C​=PC←B−1​