| 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. |
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.
