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

1.1givenL1

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

2.1step 1.1L1

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 $T (v) = \sum_{i < m} (\sum_{j < n} t_{i j} x_{j}) c_{i}$ .

3.1step 2.1L1∎

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

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

[T(v)]C=[T]BC[v]B[T(v)]C​=[T]BC​[v]B​

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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