Covector

A covector, also called a linear functional, on a vector space V over a field F is a linear map from V to F. The set of all covectors on V forms the dual space V*, which always has a linear structure and, in finite dimensions, has the same dimension as V.

In finite-dimensional spaces, a basis {e_i} of V induces a corresponding dual basis {e^i} in V*, defined by e^i(e_j) = δ^i_j. Any covector φ ∈ V* can be written as φ = ∑ φ_i e^i, with components φ_i = φ(e_i). The action on a vector v with components v^i is φ(v) = ∑ φ_i v^i. Under a change of basis, covector components transform covariantly (with the inverse transpose of the basis-change matrix), while vectors transform contravariantly.

In spaces equipped with an inner product, there is an isomorphism V ≅ V* that identifies vectors with covectors via v ↦ (w ↦ ⟨v, w⟩). In coordinates, a covector can be represented by a row vector that acts on a column vector by multiplication. Differential forms are examples of covectors in differential geometry.

Examples and applications include the differential df of a scalar field f, which is a covector at each point of a manifold, and, via a metric, the relationship between df and the gradient. Covectors are rank-1 covariant tensors; together with vectors they form higher-rank tensors and underpin the structure of the cotangent bundle in geometry.

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