# Glossary - B

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

back-substitution (with matrix notation): The backward phase of row reduction of an augmented matrix that transforms an echelon matrix into a reduced echelon matrix; used to ﬁnd the solution(s) of a system of linear equations. backward phase (of row reduction): The last part of the algorithm that reduces a matrix in echelon form to a reduced echelon form. band matrix: A matrix whose nonzero entries lie within a band along the main diagonal. barycentric coordinates (of a point p with respect to an afﬁnely independent set S D fv1 ; : : : ; vk g): The (unique) set of weights c1 ; : : : ; ck such that p D c1 v1 C ! ! ! C ck vk and c1 C ! ! ! C ck D 1. (Sometimes also called the afﬁne coordinates of p with respect to S .) basic variable: A variable in a linear system that corresponds to a pivot column in the coefﬁcient matrix. basis (for a nontrivial subspace H of a vector space V /: An indexed set B D fv1 ; : : : ; vp g in V such that: (i) B is a linearly independent set and (ii) the subspace spanned by B coincides with H , that is, H D Span fv1 ; : : : ; vp g. B-coordinates of x: basis B. best approximation: given vector. See coordinates of x relative to the the closest point in a given subspace to a bidiagonal matrix: A matrix whose nonzero entries lie on the main diagonal and on one diagonal adjacent to the main diagonal. block diagonal (matrix): A partitioned matrix A D ŒAij ! such that each block Aij is a zero matrix for i ¤ j . block matrix: See partitioned matrix. block matrix multiplication: The row–column multiplication of partitioned matrices as if the block entries were scalars. block upper triangular (matrix): A partitioned matrix A D ŒAij ! such that each block Aij is a zero matrix for i > j. boundary point of a set S in Rn : A point p such that every open ball in Rn centered at p intersects both S and the complement of S. bounded set in Rn : A set that is contained in an open ball B.0; ı/ for some ı > 0. B-matrix (for T ): A matrix ŒT !B for a linear transformation T W V ! V relative to a basis B for V , with the property that ŒT .x/!B D ŒT !B Œx!B for all x in V.
 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.