3.16.1 Tensor Covector Pullback Source Map

Tensor Covector Pullback Source Map is the underlying linear map f: V → W that supplies the mechanism by which a covector on W is converted into a covector on V through the pullback operation. Within the pullback construction (f*φ)(v) = φ(f(v)), the map f plays a specific and indispensable role distinct from that of the covector φ itself: it is the "source map," the channel through which every vector argument passes before reaching φ, and its properties, such as injectivity, surjectivity, or rank, directly determine what kind of pullback operation results.

The Role of the Source Map

Supplying the Route for Arguments

In the pullback formula (f*φ)(v) = φ(f(v)), the vector v never reaches φ directly; it is first transformed by the source map f into an element of W, and only that image is fed into φ. The source map therefore determines the entire route by which information about v reaches the covector, and different choices of source map produce entirely different pullback covectors even when φ is held fixed.

Distinguishing the Source Map from the Covector

The pullback operation depends on two independent pieces of data: the covector φ ∈ W* being pulled back, and the source map f: V → W doing the pulling. Changing φ while keeping f fixed produces a different pullback covector on V through the same route; changing f while keeping φ fixed produces a different pullback covector by altering the route itself. The term "source map" isolates this second piece of data as the object responsible for defining the domain V into which φ is being pulled.

Dependence of the Pullback on Properties of the Source Map

Injective Source Maps

If the source map f is injective, then the pullback operation f*: W* → V* is surjective: every covector on V arises as the pullback of some covector on W. Injectivity of f means distinct vectors in V map to distinct vectors in W, which allows any linear functional prescribed on V to be extended, via a choice of complement, to a functional on W that pulls back to it.

Surjective Source Maps

If the source map f is surjective, then the pullback operation f* is injective: distinct covectors on W pull back to distinct covectors on V. Surjectivity of f means every vector in W is reached by some vector in V, so a covector on W is fully determined by its values on the images of f, leaving no information lost in the pullback.

Bijective Source Maps

If the source map f is a linear isomorphism, then the pullback operation f* is also a linear isomorphism, and in this special case the pullback has a well-defined inverse operation, the pushforward (f^{-1})*, allowing covectors to be transported freely in both directions between V* and W*.

The Source Map's Matrix and the Pullback Formula

Source Map as the Transpose Factor

Relative to bases, if the source map f is represented by the matrix A, then the pullback of a covector with component row vector φ is computed as the matrix product φA, where the source map's matrix multiplies the covector's components on the right. This places the source map's matrix, not its transpose, directly into the pullback formula when covectors are treated as row vectors; equivalently, treating covectors as column vectors of coefficients gives the pullback as A^T φ, the transpose form, showing that the same operation admits both a row-vector and a transpose-matrix description depending on convention.

Rank of the Source Map Bounds the Pullback's Injectivity

The rank of the source map f controls how much information the pullback operation can distinguish: if f has rank r, the image of the pullback map f* has dimension r as well, since the rank of a matrix and the rank of its transpose coincide. A source map of low rank produces a pullback operation whose image is a correspondingly small subspace of V*, regardless of how the covector φ is chosen.

Composability of Source Maps

Chaining Source Maps

If f: V → W and g: W → U are two source maps used in sequence, the composite source map g ∘ f: V → U produces a pullback operation that equals the pullback along f applied after the pullback along g:

This reflects that pulling a covector back through a composite source map is equivalent to pulling it back through the second map first, then through the first, since the composite route through V → W → U matches the route a vector in V actually takes to reach U.

Identity as the Trivial Source Map

When the source map is the identity map id_V: V → V, the pullback operation is also the identity on V*, since routing a vector through the identity map before applying a covector changes nothing about the resulting value.

Diagrammatic Summary

The diagram labels f explicitly as the source map that carries every vector argument from V into W before the covector φ is applied, making explicit that the pullback covector f*φ is the composite φ ∘ f routed entirely through f.