# Glossary - E

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

echelon form (or row echelon form, of a matrix): An echelon matrix that is row equivalent to the given matrix. echelon matrix (or row echelon matrix): A rectangular matrix that has three properties: (1) All nonzero rows are above any row of all zeros. (2) Each leading entry of a row is in a column to the right of the leading entry of the row above it. (3) All entries in a column below a leading entry are zero. eigenfunctions (of a differential equation x0 .t/ D Ax.t/): A function x.t/ D ve "t , where v is an eigenvector of A and " is the corresponding eigenvalue. eigenspace (of A corresponding to "): The set of all solutions of Ax D "x, where " is an eigenvalue of A. Consists of the zero vector and all eigenvectors corresponding to ". eigenvalue (of A): A scalar " such that the equation Ax D "x has a solution for some nonzero vector x. eigenvector (of A): A nonzero vector x such that Ax D "x for some scalar ". eigenvector basis: A basis consisting entirely of eigenvectors of a given matrix. eigenvector decomposition (of x): An equation, x D c1 v1 C ! ! ! C cn vn , expressing x as a linear combination of eigenvectors of a matrix. elementary matrix: An invertible matrix that results by performing one elementary row operation on an identity matrix. elementary row operations: (1) (Replacement) Replace one row by the sum of itself and a multiple of another row. (2) Interchange two rows. (3) (Scaling) Multiply all entries in a row by a nonzero constant. equal vectors: Vectors in Rn whose corresponding entries are the same. equilibrium prices: A set of prices for the total output of the various sectors in an economy, such that the income of each sector exactly balances its expenses. equilibrium vector: See steady-state vector. equivalent (linear) systems: Linear systems with the same solution set. exchange model: See Leontief exchange model. existence question: Asks, “Does a solution to the system exist?” That is, “Is the system consistent?” Also, “Does a solution of Ax D b exist for all possible b?” expansion by cofactors: See cofactor expansion. explicit description (of a subspace W of Rn ): A parametric representation of W as the set of all linear combinations of a set of speciﬁed vectors. extreme point (of a convex set S ): A point p in S such that p is not in the interior of any line segment that lies in S . (That is, CONFIRMING PAGES if x, y are in S and p is on the line segment xy, then p D x or p D y.)
 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.