Glossary - F

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

factorization (of A): An equation that expresses A as a product of two or more matrices. final demand vector (or bill of final demands): The vector d in the Leontief input–output model that lists the dollar values of the goods and services demanded from the various sectors by the nonproductive part of the economy. The vector d can represent consumer demand, government consumption, surplus production, exports, or other external demand. finite-dimensional (vector space): A vector space that is spanned by a finite set of vectors. flat (in Rn ): A translate of a subspace of Rn . flexibility matrix: A matrix whose j th column gives the deflections of an elastic beam at specified points when a unit force is applied at the j th point on the beam. floating point arithmetic: Arithmetic with numbers represented as decimals ˙ :d1 ! ! ! dp # 10r , where r is an integer and the number p of digits to the right of the decimal point is usually between 8 and 16. flop: One arithmetic operation .C; "; %; =/ on two real floating point numbers. forward phase (of row reduction): The first part of the algorithm that reduces a matrix to echelon form. Fourier approximation (of order n): The closest point in the subspace of nth-order trigonometric polynomials to a given function in C Œ0; 2$!. Fourier coefficients: The weights used to make a trigonometric polynomial as a Fourier approximation to a function. the free variables (the parameters), if any. After Section 1.5, the parametric description is written in vector form.

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.
Affine combination:	A linear combination of vectors (points in Rn ) in which the sum of the weights involved is 1.
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.