Transpose of a linear map

In linear algebra, the transpose of a linear map between two vector spaces is an induced map between the dual spaces of the two vector spaces. The transpose of a linear map is often used to study the original linear map. This concept is generalized by adjoint functors.

Definition

If f : V → W is a linear map, then the transpose^[1] (or dual, or adjoint^[2]), denoted by f^∗ : W^∗ → V^∗ or by ^tf : W^∗ → V^∗, is defined to be

$f^*(\varphi) = \varphi \circ f$

for every φ ∈ W^∗. The resulting functional f^∗(φ) in V^∗ is called the pullback of φ along f.

The following identity, which characterizes the transpose,^[3] holds for all φ ∈ W^∗ and v ∈ V:

$[f^*(\varphi),\, v]_V = [\varphi,\, f(v)]_W,$

where the bracket [·,·] on the left is the duality pairing of V's dual space with V, and that on the right is the same with W.

Properties

The assignment f ↦ f^∗ produces an injective linear map between the space of linear operators from V to W and the space of linear operators from W^∗ to V^∗; this homomorphism is an isomorphism if and only if W is finite-dimensional. If V = W then the space of linear maps is an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that (fg)^∗ = g^∗f^∗. In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself. Note that one can identify (f^∗)^∗ with f using the natural injection into the double dual.

• If u : X → Y and v : Y → Z are linear maps then (v ∘ u)^∗ = u^∗ ∘ v^∗.^[4]
• If u : X → Y is a linear map, A ⊆ X, B ⊆ Y, and A° denotes the polar set of a set then^[4]
  • [u(A)]° = (u^∗)⁻¹(A°), and
  • u(A) ⊆ B implies u^∗(B°) ⊆ A°

Representation as a matrix

If the linear map f is represented by the matrix A with respect to two bases of V and W, then f^∗ is represented by the transpose matrix A^T with respect to the dual bases of W^∗ and V^∗, hence the name. Alternatively, as f is represented by A acting on the left on column vectors, f^∗ is represented by the same matrix acting on the right on row vectors. These points of view are related by the canonical inner product on Rⁿ, which identifies the space of column vectors with the dual space of row vectors.

Relation to the Hermitian adjoint

The identity that characterizes the transpose, that is, [f^∗(φ), v] = [φ, f(v)], is formally similar to the definition of the Hermitian adjoint, however, the transpose and the Hermitian adjoint are not the same map. The difference stems from the fact that transpose is defined by a bilinear form while the Hermitian adjoint is defined by a sesquilinear form. Furthermore, while the transpose can be defined on any vector space, the Hermitian adjoint is defined on Hilbert spaces.

If X and Y are Hilbert spaces and u : X → Y is a linear map then the transpose of u, which we will denote by ^tu and the Hermitian adjoint of u, which we will denote by u^∗ are related. Denote by I : X → X^∗ and J : Y → Y^∗ the canonical antilinear isometries of the Hilbert spaces X and Y onto their duals. Then u^∗ is the following composition of maps:^[5]

$Y \overset{J}{\longrightarrow} Y^* \overset{{}^{\text{t}}u}{\longrightarrow} X^* \overset{I^{-1}}{\longrightarrow} X$

Applications to functional analysis

Suppose that X and Y are topological vector spaces and that u : X → Y is a linear map, then many of u's properties are reflected in u^∗.

• If A ⊆ X and B ⊆ Y are weakly closed, convex sets containing 0, then u^∗(B°) ⊆ A° implies u(A) ⊆ B.^[4]
• The null space of u^∗ is the subspace of Y^∗ orthogonal to the range u(X) of u.^[4]
• u^∗ is injective if and only if the range u(X) of u is weakly closed.^[4]

See also

• Adjoint functors
• Hermitian adjoint
• Dual space § Transpose of a linear map

Notes

1. Treves (1999) p. 240
2. Schaefer (1999) p. 128
3. Halmos (1974, §44)
4. Schaefer (1999), pp. 129–130
5. Treves (1999) p. 488

References

• Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4
• Schaefer, Helmuth H.; Wolff, M.P. (1999). Topological Vector Spaces. GTM 3. New York: Springer-Verlag. ISBN 9780387987262.
• Trèves, François (1995). Topological Vector Spaces, Distributions and Kernels. Dover Publications. ISBN 9780486453521.