# 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. |