TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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.

In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix

Statement

Let $T : V \to W$ be a linear map between finite-dimensional inner product spaces. In orthonormal bases $E = (e_{j})$ of $V$ and $F = (f_{i})$ of $W$ ,

$[T^{*}]_{E \leftarrow F} = {\overline{[T]_{F \leftarrow E}}}^{T} .$

Over $R$ this is the transpose. The statement includes zero-sized bases.

Facts & Assumptions

Given: A map $T : V \to W$ and orthonormal bases $E = (e_{j})$ and $F = (f_{i})$ .

[L1]

The adjoint satisfies $⟨ T e_{j}, f_{i} ⟩ = ⟨ e_{j}, T^{*} f_{i} ⟩$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ , Every linear map between finite-dimensional inner product spaces has a unique adjoint).

[L2]

If $(e_{i})_{i < r}$ is an orthonormal basis, then $v = \sum_{i < r} ⟨ v, e_{i} ⟩ e_{i}$ for every vector $v$ (Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis).

[L3]

Matrix columns record the coordinates of images of basis vectors (Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases).

[L4]

Transposition interchanges matrix rows and columns, and complex conjugation is an involution (The transpose $A^{T}$ of a matrix, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Proof

technique · direct

1.1

Write $A = [T]_{F \leftarrow E}$ and $B = [T^{*}]_{E \leftarrow F}$ . Applying [L2] in $F$ expands $T e_{j} = \sum_{i} ⟨ T e_{j}, f_{i} ⟩ f_{i}$ , and applying it in $E$ expands $T^{*} f_{i} = \sum_{j} ⟨ T^{*} f_{i}, e_{j} ⟩ e_{j}$ . Since a coordinate column in a basis is unique, [L3] gives $A_{i j} = ⟨ T e_{j}, f_{i} ⟩$ and $B_{j i} = ⟨ T^{*} f_{i}, e_{j} ⟩$ .

L2L3

2.1

By [L1] and conjugate symmetry, $B_{j i} = \overline{⟨ e_{j}, T^{*} f_{i} ⟩} = \overline{⟨ T e_{j}, f_{i} ⟩} = \overline{A_{i j}}$ . Thus [L4] gives $B = {\overline{A}}^{T}$ .

step 1.1L1L4

3.1

If either basis is empty, the same entrywise identity is vacuous and identifies the unique matrix of the required size.

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 61 results over 14 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

Sheldon Axler, Linear Algebra Done Right, 4th ed., result 7.9 (standard reference, not scraped)
Sergei Treil, Linear Algebra Done Wrong, §5.5.1 (standard reference, not scraped)