Covectors

# Covectors

##### Updated: November 17, 2020

A linear mapping from a vector space to a field of scalars. In other words, a linear function which acts upon a vector resulting in a real number (scalar)

$$\alpha\,:\,\mathbf{V} \longrightarrow \mathbb{R}$$

Simplistically, covectors can be thought of as “row vectors”, or:

$$\begin{bmatrix} 1 & 2 \end{bmatrix}$$

This might look like a standard vector, which would be true in an orthonormal basis, but it is not true generally. Instead, a covector acts as a function on a standard “column vector”, e.g.

\begin{bmatrix} 1\\\
2 \end{bmatrix}

where the function’s input is a/are column vector(s) and the function’s output is a member from the $$\mathbb{R}$$ set (i.e. a scalar).

covectors have linearity (commutivity and distributivity)

Calculation of this “function” is achieved with standard matrix multiplication. For instance, if we have a covector $$\alpha$$, $$[2\,\,1]$$ and a “standard” vector $$\mathbf{v}$$, $$[x\,\,y]$$:

\begin{bmatrix} 2 & 1 \end{bmatrix} \left( \begin{bmatrix} x\\\
y \end{bmatrix} \right) =\,2x + 1y

The set of all covectors which act on a set of vectors $$\mathbf{V}$$ is known as the Dual Space. Which, itself, is a vector space. These spaces are denoted with an asterisk, e.g. $$\mathbf{V}^{\ast}$$.

Also known as:

1. One-forms
2. Linear Functionals