TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

$[T (v)]_{C} = [T]_{B}^{C} [v]_{B}$

Statement

Let $T : V \to W$ be linear, let $B$ be an ordered basis of $V$ , and let $C$ be an ordered basis of $W$ . Then for every $v \in V$ ,

$[T (v)]_{C} = [T]_{B}^{C} [v]_{B} .$

Facts & Assumptions

Given: Ordered bases $B = (b_{j})_{j < n}$ and $C = (c_{i})_{i < m}$ , a linear map $T : V \to W$ , and a vector $v \in V$ .

[L1]

The coordinate column contains the unique coefficients in the ordered-basis expansion, and the $j$ -th column of $[T]_{B}^{C}$ is $[T (b_{j})]_{C}$ (Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases).

Proof

technique · direct

1.1

Write $[v]_{B} = (x_{j})_{j < n}$ , so [L1] gives $v = \sum_{j < n} x_{j} b_{j}$ .

givenL1

2.1

By linearity, $T (v) = \sum_{j < n} x_{j} T (b_{j})$ ; writing $T (b_{j}) = \sum_{i < m} t_{i j} c_{i}$ gives $T (v) = \sum_{i < m} (\sum_{j < n} t_{i j} x_{j}) c_{i}$ .

step 1.1L1

3.1

The inner sum is the $i$ -th row-by-column entry of $[T]_{B}^{C} [v]_{B}$ , and uniqueness of $C$ -coordinates identifies this column with $[T (v)]_{C}$ .

step 2.1L1∎

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

S. Axler, Linear Algebra Done Right, 4th ed., §3C, Matrix multiplication as composition (standard reference, not scraped)
S. Schiavone, MIT 18.700 Day 9, Proposition 29 (standard reference, not scraped)

[T(v)]C=[T]BC[v]B[T(v)]C​=[T]BC​[v]B​