nLab
dual linear map

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

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Definitions

Paths and cylinders

Homotopy groups

Basic facts

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Contents

Definition

The operation VV *V \mapsto V^* of forming dual vector spaces extends to a contravariant functor.

Definition (transpose map)

The dual linear map or transpose map of a linear map A:VWA\colon V\to W, is the linear map A *=A T:W *V *A^* = A^T\colon W^*\to V^*, given by

A *(w),v=w,A(v) \langle{A^*(w), v}\rangle = \langle{w, A(v)}\rangle

for all ww in W *W^* and vv in VV.

Remark

This functor is, of course, the representable functor represented by KK as a vector space over itself (a line).

Remark

This construction is the notion of dual morphism applied in the monoidal category Vect with its tensor product monoidal structure.

Last revised on July 31, 2018 at 09:42:23. See the history of this page for a list of all contributions to it.