Dual Vector

linear-algebra

Definition

Dual Vector

A dual vector on a vector space $V$ over a field $F$ is a linear functional

$$f:V\to F.$$

So a dual vector is a linear map that takes a vector and returns a scalar.

The set of all dual vectors on $V$ forms the dual space of $V$.

Finite Dimensions

In finite-dimensional complex vector spaces, a dual vector is often written as a row vector and acts on a column vector by matrix multiplication.

In bra-ket notation, a dual vector is written as a bra $⟨x|$.