Glossary - F

All terms beginning with the letter 'F' are shown below:

Adjugate (or classical adjoint): The matrix adj A formed from a square matrix A by replacing the .i; j /-entry of A by the .i; j /-cofactor, for all i and j , and then transposing the resulting matrix.
Field:

A field is a set 𝔽 with at least two elements together with a function 𝔽 × 𝔽 ⇾ 𝔽 called addition, denoted (a, b) → a + b, and a function 𝔽 × 𝔽 ⇾ 𝔽 called multiplication, denoted (a, b) → ab, which satisfy the following axioms:

  1. Commutativity of addition and multiplication: a + b = b + a, and a ⋅ b = b ⋅ a.
  2. Associativity of addition and multiplication: a + (b + c) = (a + b) + c, and a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c.
  3. Additive and multiplicative identity: there exist two different elements 0 and 1 in 𝔽 such that a + 0 = a and a ⋅ 1 = a.
  4. Additive inverses: for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.
  5. Multiplicative inverses: for every a ≠ 0 in F, there exists an element in 𝔽, denoted by a−1 or 1/a, called the multiplicative inverse of a, such that a ⋅ a−1 = 1.
  6. Distributivity of multiplication over addition: a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c).

Affine dependence relation:

An equation of the form c1 v1 C ! ! ! C cp vp D 0, where the weights c1 ; : : : ; cp are not all zero, and c1 C ! ! ! C cp D 0.
Affine hull (or affine span) of a set S:

The set of all affine combinations of points in S , denoted by aff S.
Affinely dependent set:

A set fv1 ; : : : ; vp g in Rn such that there are real numbers c1 ; : : : ; cp , not all zero, such that c1 C ! ! ! C cp D 0 and c1 v1 C ! ! ! C cp vp D 0.
Affinely independent set:

A set fv1 ; : : : ; vp g in Rn that is not affinely dependent.
Affine set (or affine subset):

A set S of points such that if p and q are in S , then .1 " t/p C t q 2 S for each real number t.
Affine transformation:

A mapping T W Rn ! Rm of the form T .x/ D Ax C b, with A an m # n matrix and b in Rm.
Algebraic multiplicity:

The multiplicity of an eigenvalue as a root of the characteristic equation.
Angle (between nonzero vectors u and v in R2 or R3/:

The angle # between the two directed line segments from the origin to the points u and v. Related to the scalar product by u ! v D kuk kvk cos #
Associative law of multiplication:

A.BC/ D .AB/C , for all A, B, C.
attractor (of a dynamical system in R2):

The origin when all trajectories tend toward 0.
Augmented matrix:

A matrix made up of a coefficient matrix for a linear system and one or more columns to the right. Each extra column contains the constants from the right side of a system with the given coefficient matrix.
Auxiliary equation:

A polynomial equation in a variable r, created from the coefficients of a homogeneous difference equation.