Glossary - G

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

Givens rotation: A linear transformation from Rn to Rn used in computer programs to create zero entries in a vector (usually a column of a matrix). Gram matrix (of A): The matrix ATA. Gram–Schmidt process: An algorithm for producing an orthogonal or orthonormal basis for a subspace that is spanned by a given set of vectors.
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.