In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

 concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace

Field trace

In mathematics, the field trace is a function defined with respect to a finite field extension L/K. It is a K-linear map from L to K...

. This requires the property that the field trace Tr_L/K provides a non-degenerate quadratic form

Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

 over K. This can be guaranteed if the extension is separable

Separable extension

In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...

; it is automatically true if K is a perfect field

Perfect field

In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...

, and hence in the cases where K is finite, or of characteristic zero.

A dual basis isn't a concrete basis

Basis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

 like the polynomial basis

Polynomial basis

In mathematics, the polynomial basis is a basis for finite extensions of finite fields.Let α ∈ GF be the root of a primitive polynomial of degree m over GF...

 or the normal basis

Normal basis

In mathematics, a normal basis in field theory is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis...

; rather it provides a way of using a second basis for computations.

Consider two bases for elements in a finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

, GF(p^m):


and


then B₂ can be considered a dual basis of B₁ provided


Here the trace

Field trace

In mathematics, the field trace is a function defined with respect to a finite field extension L/K. It is a K-linear map from L to K...

 of a value in GF(p^m) can be calculated as follows:


Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with a multiplication by the multiplicative identity (usually 1).