# 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)

\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**:

- One-forms
- Linear Functionals