Alphabeta MathHow 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.
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The transpose or algebraic adjoint , , of a linear map
Definition
Let be linear. Its transpose, or algebraic adjoint, is the map
The composite is linear, so it belongs to . This definition is algebraic and does not use an inner product.
Depends on
Used by
- Transpose is linear, sends identities to identities, and reverses composition: (S∘ T)^*=T^*∘ S^* Proposition
- Assuming choice, ker T^*=(imT)^∘ and imT^*=(ker T)^∘; in finite dimensions rankT^*=rankT Theorem
- Dual and Hom transition functions define smooth bundles Theorem
- In dual bases, the matrix of T^* is the transpose of the matrix of T Theorem
Dependency tree · two levels
4 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
- H. Pinkham, Linear Algebra, §§6.4–6.6 (standard reference, not scraped)