Linear form

A Linear map where the image is the underlying field of the vector space, e.g. ${\mathbb{R}}^n\to \mathbb{R}$.

The set of all linear forms over a vector space forms another vector space called the dual space.

For the typical case of a linear form over ${\mathbb{R}}^n$, the form can be seen just as a row vector with n elements, the full form being specified by the value of each of the basis vectors.

The dual space of a vector space $V$, sometimes denoted $V^{\ast }$, is the vector space of all linear forms over $V$ with the obvious addition and scalar multiplication operations defined.

Since a linear form is completely determined by how it acts on a basis, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the $V$, and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.

Infinite dimension seems more interesting however, see: en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278#Infinite-dimensional_case

One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space $V$.

Dual vectors are the members of a dual space.

In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis:The dual basis vectors $e^i$ are defined to "pick the corresponding coordinate" out of elements of V. E.g.:By expanding into the basis, we can put this more succinctly with the Kronecker delta as:

Note that in Einstein notation, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector $f$ as:

In the context of quantum mechanics, the bra notation is also used for dual vectors.