Linear Algebra

Frage	Antworten
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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	Erstellt von Kimberly Pruitt vor etwa 8 Jahre