Covectors

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)

\begin{equation} \alpha\,:\,\mathbf{V} \longrightarrow \mathbb{R} \end{equation}

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

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

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

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

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