Covectors

Covectors are linear functionals on a vector space, elements of the dual space V*, which comprises all linear maps from V to the underlying field F. A covector assigns to each vector a scalar, preserving addition and scalar multiplication.

In finite dimensions, every basis {e_i} of V has a corresponding dual basis {e^i} in V*, defined by e^i(e_j) = δ^i_j. Any covector ω can be written as ω = Σ_i ω_i e^i, and for a vector v = Σ_i v_i e_i, the action is ω(v) = Σ_i ω_i v_i. In matrix terms, vectors are represented by column vectors and covectors by row vectors, so ω(v) = ω^T v.

Under a change of basis, covectors transform with the inverse transpose: ω' = (P^{-1})^T ω, ensuring the scalar ω(v) remains unchanged for corresponding transformed vectors and covectors.

If V carries an inner product ⟨·,·⟩, the inner product induces an isomorphism V → V* by w ↦ ω_w with ω_w(v) = ⟨w, v⟩. Thus, via this identification, covectors can be viewed as vectors themselves in the same space. In Euclidean space with the standard dot product, covectors correspond to row vectors acting on column vectors by dot product.

In differential geometry, a covector at a point (a 1-form) is a linear functional on the tangent space at that point, and a covector field assigns a covector smoothly to each point on a manifold.

Examples: In R^2 with the standard basis, a covector ω = a e^1 + b e^2 acts on v = (v1, v2) as ω(v) = a v1 + b v2. Covectors provide a natural way to evaluate vectors and to formulate duality in linear algebra and geometry.