6381443
Description
Flashcards by Kimberly Pruitt, updated more than 1 year ago
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Created by Kimberly Pruitt
about 8 years ago
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| Question | Answer |
| 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. |
| BASIC VARIABLE | A variable in a LINEAR SYSTEM that corresponds to a PIVOT COLUMN in the COEFFICIENT MATRIX. |
| CODOMAIN | (of a transformation T: Rn --> Rm): The set Rm that contains the RANGE of T. In general, if T maps a VECTOR SPACE V into a VECTOR SPACE W, then W is called the CODOMAIN of T. |
| COEFFICIENT MATRIX | A MATRIX whose entries are the coefficients of a SYSTEM OF LINEAR EQUATIONS. |
| COLUMN VECTOR | A MATRIX with only one column, or a single column of a MATRIX that has several columns. |
| CONSISTENT LINEAR SYSTEM | A LINEAR SYSTEM with at least one solution. |
| DILATION | A MAPPING x |--> rx for some SCALAR, with 1 < r. |
| DOMAIN | (of a transformation T): The set of all VECTORS x for which T(x) s defined. |
| ECHELON FORM | An ECHELON MATRIX that is ROW EQUIVALENT to the given MATRIX. |
| 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. |
| EQUIVALENT (LINEAR) SYSTEMS | LINEAR SYSTEMS with the same SOLUTION SET. |
| FLOP | One arithmetic operation (+, -, *, /) on two real floating point numbers. |
| FREE VARIABLE | Any variable in a LINEAR SYSTEM that is not a BASIC VARIABLE. |
| HOMOGENOUS EQUATION | An equation of the form Ax=0, possibly written as a VECTOR EQUATION or as a SYSTEM OF LINEAR EQUATIONS. |
| IDENTITY MATRIX | (denoted by I or In): A square MATRIX with ones on the diagonal and zeros elsewhere. |
| IMAGE | (of a VECTOR x under a TRNSFORMATION T): The VECTOR T(x) assigned to x by T. |
| INCONSISTENT LINEAR SYSTEM | A LINEAR SYSTEM with no solution. |
| INTERPOLATING POLYNOMIAL | A polynomial whose graph passes through every point in a set of data points in R2. |
| LEADING ENTRY | The leftmost nonzero entry in a row of a MATRIX. |
| LINEAR COMBINATION | A sum of SCALAR MULTIPLES of VECTORS. The SCALARS are called the WEIGHTS. |
| LINEAR DEPENDENCE RELATION | A HOMOGENOUS VECTOR EQUATION where the WEIGHTS are all specified and at least one WEIGHT is nonzero. |
| LINEAR EQUATION | (in the variables x1, . . ., xn): An equation that can be written in the form a1x1 + a2x2 + . . . + anxn = b, where b and the coefficients a1, . . ., an are real or complex numbers |
| LINEAR SYSTEM | A collection of one or more LINEAR EQUATIONS involving the same variables, say x1, . . ., xn. |
| LINEARLY DEPENDENT | An indexed set {v1, . . ., vp} with the property that there exist WEIGHTS c1, . . ., cp, not all zero, such that c1v1 + . . . + cpvp = 0. That is, the VECTOR EQUATION c1v1 + c2v2 + . . . + cpvp = 0 has a NONTRIVIAL SOLUTION. |
| LINEARLY INDEPENDENT | An indexed set {v1, . . ., vp} with the property that the VECTOR EQUATION c1v1 + c2v2 + . . . + cpvp = 0 has only the TRIVIAL SOLUTION, c1 = . . . = cp = 0. |
| MAPPING | See TRANSFORMATION |
| MATRIX | A rectangular array of numbers |
| MATRIX EQUATION | An equation that involves at least one MATRIX; for instance, Ax=b. |
| MATRIX SIZE | Two numbers, written in the form m x n, that specify the number of rows (m) and columns (n) in the MATRIX. |
| NONTRIVIAL SOLUTION | A nonzero solution of a HOMOGENOUS EQUATION or systems of HOMOGENOUS EQUATIONS. |
| NONHOMOGENOUS EQUATION | An equation of the form Ax = b with b not equal to 0, possibly written as a VECTOR EQUATION or as a SYSTEM OF LINEAR EQUATIONS. |
| PARAMETRIC EQUATION OF A LINE | An equation of the form x = p + tv (t in R). |
| PARAMETRIC EQUATION OF A PLANE | An equation of the form x = p + su + tv (s, t in R), with u and v LINEARLY INDEPENDENT. |
| PIVOT | A nonzero number that either is used in a PIVOT POSITION to create zeros through row operations or is changed into a leading 1, which in turn is used t create zeros. |
| PIVOT COLUMN | A column that contains a PIVOT POSITION. |
| PIVOT POSITION | A position in a MATRIX A that corresponds to a LEADING ENTRY in an ECHELON FORM of A. |
| RANGE | (of a linear TRANSFORMATION T): The set of all VECTORS of the form T(x) for some x in the DOMAIN of T. |
| REDUCED ECHELON FORM | A REDUCED ECHELON MATRIX that is ROW EQUIVALENT to a given MATRIX. |
| REDUCED ECHELON MATRIX | A rectangular MATRIX in ECHELON FORM that has these additional properties: The LEADING ENTRY in each nonzero row is 1, and each leading 1 is the only nonzero entry in its column. |
| ROW EQUIVALENT | Two matrices for which there exists a (finite) sequence of row operations that transform one MATRIX into the other. |
| SCALAR | A (real) number used to multiply either a VECTOR or a MATRIX. |
| SCALAR MULTIPLES | (of u by c): The VECTOR cu obtained by multiplying each entry in u by c. |
| SOLUTION SET | The set of all possible solutions of a LINEAR SYSTEM. The SOLUTION SET is empty when the LINEAR SYSTEM is INCONSISTENT. |
| SPAN | The set of all LINEAR COMBINATIONS of v1, . . ., vp. Also, the subspace spanned (or generated) by v1, . . ., vp. |
| SUBSPACE | A subset H of some VECTOR SPACE V such that H has these properties: (1) the zero VECTOR of V is in H; (2) H is closed under VECTOR addition; and (3) H is closed under multiplication by SCALARS. |
| SYSTEM OF LINEAR EQUATIONS | A collection of one or more LINEAR EQUATIONS involving the same set of variables, say, x1, . . ., xn. |
| TRANSFORMATION | A rule that assigns to each VECTOR x in Rn a unique VECTOR T(x) in Rm. Notation T: Rn -->Rm. Also T: V --> W denotes a rule that assigns to each x in V a unique VECTOR T(x) in W. |
| TRIVIAL SOLUTION | The solution x=0 of a HOMOGENOUS EQUATION Ax=0. |
| VECTOR | A list of numbers; a MATRIX with only one column. In general, any element of a VECTOR SPACE |
| VECTOR EQUATION | An equation involving a LINEAR COMBINATION of VECTORS with undetermined WEIGHTS. |
| VECTOR SPACE | A set of objects, called VECTORS, on which two operations are defined, called addition and multiplication by SCALARS. Ten axioms (rules) must be satisfied. |
| WEIGHTS | The SCALARS used in a LINEAR COMBINATION. |
| ZERO VECTOR | The unique VECTOR, denoted by 0, such that u + = u for all u. In Rn, 0 is the VECTOR whose entries are all zeros. |
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