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)

α:V⟶R

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

[12]

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.

[1 2]

where the function’s input is a/are column vector(s) and the function’s output is a member from the 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 α, [21] and a “standard” vector v, [xy]:

[21]([x y])=2x+1y

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

Also known as:

1. One-forms
2. Linear Functionals