Transpose of a linear map

Key takeaways

Transpose of a linear map belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Transpose of a linear map to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Transpose of a linear map from memory before moving on to harder problems.

Reference excerpt

In linear algebra and functional analysis, the transpose or algebraic adjoint of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose is often used to study the original linear map. This concept is generalised by adjoint functors.

Definition

Let X # {\displaystyle X^{\#}} denote the algebraic dual space of a vector space ⁠ X {\displaystyle X} ⁠. Let X {\displaystyle X} and Y {\displaystyle Y} be vector spaces over the same field ⁠ K {\displaystyle {\mathcal {K}}} ⁠. If u : X → Y {\displaystyle u:X\to Y} is a linear map, then its algebraic adjoint, or dual, is the map

# u : Y # → X # {\displaystyle {}^{\#\!}u:Y^{\#}\to X^{\#}} defined by ⁠ f ↦ f ∘ u {\displaystyle f\mapsto f\circ u} ⁠. The resulting functional

# u ( f ) := f ∘ u {\displaystyle {}^{\#\!}u(f):=f\circ u} is called the pullback of f {\displaystyle f} by ⁠ u {\displaystyle u} ⁠. The continuous dual space of a topological vector space (TVS) X {\displaystyle X} is denoted by ⁠ X ′ {\displaystyle X^{\prime }} ⁠. If X {\displaystyle X} and Y {\displaystyle Y} are TVSs then a linear map u : X → Y {\displaystyle u:X\to Y} is weakly continuous if and only if ⁠

# u ( Y ′ ) ⊆ X ′ {\displaystyle {}^{\#\!}u\left(Y^{\prime }\right)\subseteq X^{\prime }} ⁠, in which case we let

t u : Y ′ → X ′ {\displaystyle {}^{\text{t}}\!u:Y^{\prime }\to X^{\prime }} denote the restriction of

# u {\displaystyle {}^{\#\!}u} to ⁠ Y ′ {\displaystyle Y^{\prime }} ⁠. The map

t u {\displaystyle {}^{\text{t}}\!u} is called the transpose or algebraic adjoint of ⁠ u {\displaystyle u} ⁠. The following identity characterizes the transpose of ⁠ u {\displaystyle u} ⁠:

⟨

t u ( f ) , x ⟩ = ⟨ f , u ( x ) ⟩ for all f ∈ Y ′ and x ∈ X , {\displaystyle \left\langle {}^{\text{t}}\!u(f),x\right\rangle =\left\langle f,u(x)\right\rangle \quad {\text{ for all }}f\in Y^{\prime }{\text{ and }}x\in X,}

where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \left\langle \cdot ,\cdot \right\rangle } is the natural pairing defined by ⁠ 1 {\displaystyle {1}} ⁠.

Properties The assignment u ↦

t u {\displaystyle u\mapsto {}^{\text{t}}\!u} produces an injective linear map between the space of linear operators from X {\displaystyle X} to Y {\displaystyle Y} and the space of linear operators from Y # {\displaystyle Y^{\#}} to ⁠ X # {\displaystyle X^{\#}} ⁠. If X = Y {\displaystyle X=Y} then the space of linear maps is an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that ⁠

t ( u v ) =

t v

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Transpose of a linear map

Start with the simplest possible case. Write down what Transpose of a linear map claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Transpose of a linear map before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Transpose of a linear map ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Transpose of a linear map

In research: Transpose of a linear map appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Transpose of a linear map in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Transpose of a linear map is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functional analysis, Linear algebra, Linear functionals, so understanding it makes those chapters shorter.
In everyday life: Look for Transpose of a linear map outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Transpose of a linear map in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Transpose of a linear map means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Transpose of a linear map out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Transpose of a linear map in simple terms?

In linear algebra and functional analysis, the transpose or algebraic adjoint of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose is often used to study the original linear map.

Why does Transpose of a linear map matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Transpose of a linear map?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Transpose of a linear map.