1.2.15 Covector Definition

Covector Definition is the characterization of a covector as an element of the dual space of a vector space, equivalently, as a linear functional on that vector space, distinguished from an ordinary vector by the way its numerical components transform under a change of basis. It supplies the name given to the fundamental objects of the dual space, and it is the covariant counterpart to the vector, together forming the two basic kinds of rank-one tensors from which every higher-rank tensor is ultimately constructed.

Covectors as Elements of the Dual Space

A covector is, by definition, a member of the dual space associated with a given vector space, which means it is itself a linear functional: a linear map that accepts a vector as input and returns a scalar as output. Because the dual space is itself a vector space, covectors can be added together and scaled by elements of the underlying field, just as ordinary vectors can, and every property that holds for vectors in general — bases, linear independence, coordinates — applies equally to covectors, since they inhabit a vector space in their own right.

Terminology

The term covector is used interchangeably, depending on context and discipline, with dual vector and one-form. The term dual vector emphasizes the covector's status as an element of the dual space, paired naturally with the vectors of the original space. The term one-form is more common in differential geometry, where covectors arise as the basic building blocks of differential forms, generalizing the covector concept to fields of covectors varying smoothly over a manifold. All three terms refer to the same underlying algebraic object.

The Natural Pairing with Vectors

The defining feature of a covector is its ability to act on a vector to produce a scalar, an operation called the natural pairing between the vector space and its dual. This pairing is bilinear: it is linear in the vector argument, since a covector is a linear functional, and linear in the covector argument, since the addition and scalar multiplication of covectors are defined precisely so that this holds.

The expression above denotes the natural pairing of a covector with a vector, producing a scalar equal to the result of applying the covector, as a linear functional, to the vector.

Components of a Covector

Once a basis is chosen for the original vector space, and the corresponding dual basis is constructed for the dual space, a covector can be expressed as a linear combination of dual basis elements, and the coefficients of this expansion are the components of the covector relative to the chosen basis. By convention, these components are written with a lower index, called a covariant index, distinguishing them from the upper, contravariant indices used for vector components. This indexing convention directly reflects how covector components transform under a change of basis: they transform using the same coefficients that describe how the original basis vectors change, rather than the inverse coefficients used for vector components.

Covectors Within Tensor Algebra

Within the general framework of tensor algebra, a covector is precisely a tensor of type (0, 1): a tensor with zero contravariant indices and one covariant index. Every mixed tensor of higher rank can be understood as being built, through the tensor product, from some combination of vectors and covectors, and every covariant index that a tensor carries corresponds to one covector-like slot in the multilinear map that the tensor represents. Recognizing a covector as this specific, simplest case of a tensor is what allows the general theory of tensors to subsume vectors, covectors, linear maps, and bilinear forms as particular instances of a single unified construction.