Lagmental Vicfred

The Dual Map Reverses a Linear Arrow by Vicfred

tags: mathematics, vector spaces, linear maps, dual spaces

Precomposition sends functionals on W back to functionals on V. The point is to make the formal expression readable enough to audit line by line.

Statement

A linear map \(T:V\to W\) is organized by its kernel \(\ker T\) and image \(\operatorname{im}T\). Quotients and duals express the same information without choosing bases.

$$ T^\ast:W^\ast\to V^\ast,\qquad T^\ast\lambda=\lambda\circ T $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \ker T^\ast=(\operatorname{im}T)^\circ,\qquad\operatorname{im}T^\ast=(\ker T)^\circ $$

Worked algebra

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ [T^\ast]_{\mathcal C^\ast\leftarrow\mathcal B^\ast}=[T]_{\mathcal C\leftarrow\mathcal B}^{\mathsf T} $$

Conceptual compression

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad T^\ast:W^\ast\to V^\ast,\qquad T^\ast\lambda=\lambda\circ T,\\[5pt] \mathsf{C}\;&:\quad \ker T^\ast=(\operatorname{im}T)^\circ,\qquad\operatorname{im}T^\ast=(\ker T)^\circ. \end{aligned} $$

Caveat

Dimension formulas below assume finite-dimensional spaces. Infinite-dimensional vector spaces require cardinal arithmetic and may not identify naturally with their double duals.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \ker T^\ast=(\operatorname{im}T)^\circ,\qquad\operatorname{im}T^\ast=(\ker T)^\circ \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Thu 17 November 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.