DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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The transpose or algebraic adjoint $T^{} : W^{} \to V^{}$ , $T^{} (g) = g \circ T$ , of a linear map $T : V \to W$

Definition

Let $T : V \to W$ be linear. Its transpose, or algebraic adjoint, is the map

$T^{*} : W^{*} ⟶ V^{*}, T^{*} (g) : = g \circ T .$

The composite $g \circ T$ is linear, so it belongs to $V^{*}$ . This definition is algebraic and does not use an inner product.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 5 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

H. Pinkham, Linear Algebra, §§6.4–6.6 (standard reference, not scraped)

The transpose or algebraic adjoint T∗:W∗→V∗T∗:W∗→V∗, T∗(g)=g∘TT∗(g)=g∘T, of a linear map T:V→WT:V→W