Transpose of a linear map

In the context of linear algebra, the transpose of a linear map is a fundamental concept that relates to how linear transformations interact with dual spaces. ### Definition Let \( T: V \to W \) be a linear map between two finite-dimensional vector spaces \( V \) and \( W \).

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T : V \to W

T^{*} : W^{*} \to V^{*}

φ

on the output space to the input measurement

T^{*} (φ) = φ \circ T

. It reverses direction because a measurement of the output can be evaluated on an input by first applying

T

(T^{*} φ) (v) = φ (T v) .

For

T (x, y) = (x + y, 2 y)

and

φ (u, v) = 2 u - v

, the input measurement is

T^{*} φ (x, y) = 2 (x + y) - 2 y = 2 x

. The output measurement ignores the part of this transformation contributed by

y

T

A

in specified bases, a row representing

φ

changes to

ℓ A

. Storing the coefficients of a linear functional as a column in the corresponding dual basis gives

A^{T} ℓ^{T}

, explaining why the transpose of a linear map is represented by the matrix transpose. This algebraic operation uses no inner product; it agrees with a matrix transpose even over the complex numbers, rather than automatically introducing complex conjugation. The adjoint operator is defined using an inner product and is a different construction. ^[ref].

ker T^{*} = (im T)^{\circ}, im T^{*} = (ker T)^{\circ} .