A vector can be transformed into a scalar by a linear measurement. Such a linear measurement is called a covector or linear functional.

If $V$ is a vector space over the real numbers, a covector is a linear map

$$\alpha:V\to\mathbb R.$$

The set of all covectors on $V$ forms the dual space $V^*$.

Linearity

For vectors $\mathbf u,\mathbf v$ and scalar $c$,

$$\alpha(\mathbf u+\mathbf v)=\alpha(\mathbf u)+\alpha(\mathbf v),$$

$$\alpha(c\mathbf u)=c\alpha(\mathbf u).$$

Components

After choosing a basis, a vector is represented by a column of components while a covector can be represented by coefficients that act on those components to produce a scalar.

The numerical representation depends on the basis, while the underlying linear functional does not.

Why distinguish vectors and covectors

In Euclidean spaces an inner product lets vectors and covectors be identified conveniently, which can hide the distinction. In more general coordinate systems and tensor calculus, keeping the two transformation behaviours separate makes the structure clearer.

Covectors provide the simplest example of a mathematical object whose components change with coordinates while the object itself remains the same.