Linear algebra vocab

algebraic multiplicity

the multiplicity of an eigenvalue as a root of the characteristic equation

associative law of multiplication

A(BC) = (AB)C

augmented matrix

a matrix made up of a coefficient matrix for a linear system and one or more columns to the right

auxiliary equation

a polynomial equation in a variable r, created from the coefficients of a homogeneous difference equation

basic variable

a variable in a linear system that corresponds to a pivot column in the coefficient matrix

basis

an indexed set B={v1.....vp} in V such that: B is a linearly independent set and H = span{v1...vp}

change of coordinate matrix

a matrix P C

characteristic equation

det(A - lambda I)=0

characteristic polynomial

det(A - labda I)

codomain

the set Rm that contains the range of T, if T maps a vector space V into a vector space W, W is the codomain

coefficient matrix

a matrix whose entries are the coefficients of a system of linear equations

cofactor expansion

a formula for det A using cofactors associated with one row or one column

column space

the set Col A of all linear combinations of the columns of A

column vector

a matrix with only one column

commuting matrices

AB = BA

composition of linear transformations

a mapping produced by applying two or more linear transformations in succession

consistent linear system

a linear system with at least one solution

coordinate mapping

a mapping that associates to each x in V its coordinate vector [x]B

coordinates of x relative to the basis B = {b1....bn}

The weights c1, ..., cn in the equation x = c1b1 + ..... + cnbn

coordinate vector of x relative to B

The vector [x]B whose entries are the coordinates of x relative to the basis B

determinant

ad-bc, real number associatied with a square matrix

diagonal entries

entries having equal row and column indices

diagonalizable

a matrix that can be written in factored form as PDP^-1 where D is the diagonal matrix and P the invertible matrix

diagonal matrix

a square matrix whose entries not on the main diagonal are all zero

dimension

of a subspace: the number of vectors in a basis for S written as dim S

of a vector space: the number of vectors in a basis for V, written dim V

distributive laws

A(B+C) = AB + AC

(B+C)A = BA + CA

domain

the set of all vectors x for which T(x) is defined

eigenspace

the set of all solutions of Ax = lamda(x) where lambda is an eigenvalue, consists of all eigenvectors and the zero vector

eigenvalue

a scalar lambda such that the equation Ax=lambda(x) has a solution for some nonzero vector x

eigenvector

a nonzero vector x such that Ax = lambda(x) for some scalar lambda

eigenvector basis

eigenvectors forming basis for Rn (invertible); necessary for nxn A to be diagonalizable

eigenvector decomposition

an equation x = c1v1+.....cnvn expressing x as a linear combination of eigenvectors of a matrix

equal vectors

vectors in Rn whose corresponding entries are the same

equivalent systems

linear systems with the same solution set

finite dimensional

a vector space that is spanned by a finite set of vectors

free variable

any variable in a linear system that is not a basic variable

full rank

an m x n matrix whose rank is the smaller of m and n

fundamental set of solutions

a basis for the set of all solutions of a homogeneous linear difference or difference equation

fundamental subspaces

the null space and column space of A, and the null space and column space of A^T, with Col A^T commonly called the row space of A

homogenous equation

an equation of the form Ax=0 possibly written as a vector equation or as a system of linear equaitons

image

the vector T(x) assigned to x by T

inconsistent linear system

a linear system with no solution

infinite dimensional

a nonzero vector space V that has no finite basis

inverse

an n x n matrix A^-1 such that AA^-1 = A^-1A = I

invertible matrix

a square matrix that possesses an inverse

isomorphic vector spaces

two vector spaces V and W for which there is a one-to-one linear transformation T that maps V onto W

isomorphism

a one-to-one linear mapping from one vector space onto another

kernel

the set of x in V such that T(x) = 0

left inverse

any rectangular matrix C such that CA = I

linear combination

a sum of scalar multiples of vectors, the scalars are called the weights

linear dependence relation

a homogeneous vector equation where the weights are all specified and at least one weight is nonzero

linear equation

an equation that can be written in the form a1x1+ a2x2 + ..... anxn = b where b and the coefficients are real numbers

linearly dependent

an indexed set {v1, .. , vp} with the property that there exist weights c1, ... , cp, not all zero, such that c1v1 + c2v2 + ... + cpvp = 0. That is, the vector equation c1v1 + c1v2 + ... + cpvp = 0 has a non trivial solution

linearly independent

an indexed set {v1, ..., vp} with the property that the vector equation c1v1 + c2v2 + ... + cnvn = 0 has only the trivial solution

linear system

a collection of one or more linear equations involving the same variables, x1....xn

linear transformation

1. T (u + v) = T (u) + T (v) for all u, v, in the domain of T; 2. T (cu) = cT(u) for all u and scalars c

main diagonal

the entries with equal row and column indices

matrix equaiton

Ax = b

matrix for T relative to bases B and C

M = [[T (b1)]C [T(b2)]C ... [T (bn) ]C]; a matrix representation of T

matrix transformation

a mapping x |-> Ax where A is an m x n matrix and x represents any vector in Rn

minimal spanning set

a set B that spans H and has the property that if one of the elements of is removed from B then the new set does not span H

non homogenous equation

Ax = b with b not equal to 0

nonsingular

an invertible matrix

nontrivial solution

a nonzero solution of a homogeneous equation or system of homogeneous equations

null space

The null space of an mxn matrix A, written nul A, is the set of all solutions of the homogeneous equation Ax=0; that is Nul A={x:x is in Rn and Ax=0}

parametric equation of line

x = p + tv

pivot

a nonzero number that either is used in a pivot position to create zeros through row operations

product Ax

the linear combination of the columns of A using the corresponding entries in x as weights

range

the set of all vectors of the form T(x) for some x in the domain of T

rank

the dimension of the column space of A, rank A

row space

the set Row A of all linear combinations of the vectors formed from the rows of A, Col A^T

set spanned by {v1....vp}

the set Span{v1.....vp}

singular

a square matrix that has no inverse

solution set

the set of all possible solutions of a linear system, set is empty when the linear system is inconsistent

span{v1....vp}

the set of all linear combinations of v1.....vp

spanning set

any set {v1....vp} in H such that H = span {v1.....vp}

standard basis

the basis E = {e1....en}

standard matrix

the matrix A such that T(x) = Ax for all x in the domain T

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

set of two or more equations with two or more variables

transformation function

a rule that assigns to each vector x in Rn a unique vector T(x) in Rm

transpose

an n x m matrix A^T whose columns are the corresponding rows of the m x n matrix A

vector

a list of numbers, matrix with only one column

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, ten axioms

zero subspace

the subspace {0} consisting of only the zero vector

zero vector

unique vector such that the entries are all zero