INDEX

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A

addition of vectors

adjugate of a matrix

algebraic multiplicity of an eigenvalue

associated homogeneous system (def:asshomsys)

augmented matrix

B

back substitution

basis (def:basis)

block matrices

box product

C

change-of-basis matrix (def:matlintransgenera)

characteristic equation (def:chareqcharpoly)

characteristic polynomial

Cholesky factorization

closed under addition (def:closedunderaddition)

closed under scalar multiplication (def:closedunderscalarmult)

codomain of a linear transformation

coefficient matrix

cofactor expansion

column matrix (vector)

column space of a matrix (def:colspace)

composition of linear transformations (def:compoflintrans)

consistent system

convergence

coordinate vector with respect to a basis (in ) (abstract vector spaces)

Cramer’s Rule

cross product (def:crossproduct)

D

determinant () (def:twodetcrossprod)

determinant () (def:threedetcrossprod)

diagonal matrix

diagonalizable matrix (def:diagonalizable)

where is a diagonal matrix. In other words, a matrix is diagonalizable if it is similar to a diagonal matrix, .

dimension (def:dimension) (also see def:dimensionabstract)

Direction vector

Distance between points in (form:distRn)

Distance between point and line

divergence

domain of a linear transformation

dominant eigenvalue (def:dominant ew,ev)

Any corresponding eigenvector is called a dominant eigenvector of .

dot product (def:dotproduct)

E

eigenspace (def:eigspace)

eigenvalue (def:eigen)

eigenvalue decomposition (def:eigdecomposition)

eigenvector (def:eigen)

elementary matrix (def:elemmatrix)

elementary row operations (def:elemrowops)

equivalence relation

equivalent linear systems (def:equivsystems)

F

free variable

fundamental subspaces of a matrix

G

Gaussian elimination (def:GaussianElimination)

Gauss-Jordan elimination (def:GaussJordanElimination)

Gauss-Seidel method

geometric multiplicity of an eigenvalue (def:geommulteig)

Gerschgorin’s Disk Theorem

Gram-Schmidt Process

H

homogeneous system (def:homogeneous)

hyperplane

I

identity matrix

identity transformation (def:idtransonrn)

image of a linear transformation (def:imageofT)

inconsistent system

inner product (def:innerproductspace)

inner product space

inverse of a linear transformation (def:inverseoflintrans)

inverse of a square matrix (def:matinverse)

isomorphism (def:isomorphism)

iterative methods

J

Jacobi’s method

K

kernel of a linear transformation (def:kernel)

L

Laplace Expansion Theorem (th:laplace1)

leading entry (leading 1) (def:leadentry)

leading variable

linear combination of vectors (def:lincomb)

linear equation (def:lineq)

linear transformation (def:lin) (also see Linear Transformations of Abstract Vector Spaces)

linearly dependent vectors (def:linearindependence)

is the trivial solution .

linearly independent vectors (def:linearindependence)

is the trivial solution .

lower triangular matrix

LU factorization

M

Magnitude of a vector (def:normrn)

main diagonal

matrix

matrix addition (def:additionofmatrices)

matrix equality (def:equalityofmatrices)

matrix multiplication (by a matrix) (def:matmatproduct)

matrix multiplication (by a scalar) (def:scalarmultofmatrices)

matrix multiplication (by a vector) (def:matrixvectormult)

matrix of a linear transformation with respect to the given bases (def:matlintransgenera)

minor of a square matrix

N

norm (def:030438)

nonsingular matrix (def:nonsingularmatrix)

normal vector

null space of a matrix (def:nullspace)

nullity of a linear transformation (def:nullityT)

nullity of a matrix (def:matrixnullity)

O

one-to-one (def:onetoone)

onto (def:onto)

ordered basis

orthogonal basis

orthogonal complement of a subspace of (def:023776)

Orthogonal Decomposition Theorem (th:OrthoDecomp)

orthogonal matrix (def:orthogonal matrices)

Orthogonal projection onto a subspace of (def:projOntoSubspace)

is called the orthogonal projection of onto .

Orthogonal projection onto a vector (def:projection)

orthogonal set of vectors (orthset)

Orthogonal vectors (def:orthovectors)

orthogonally diagonalizable matrix (def:orthDiag)

orthonormal basis

orthonormal set of vectors (orthset)

P

parametric equation of a line (form:paramlinend)

particular solution

partitioned matrices (block multiplication)

permutation matrix

pivot

positive definite matrix (def:024811)

power method (and its variants)

Q

QR factorization (def:QR-factorization)

R

rank of a linear transformation (def:rankofT)

rank of a matrix (def:rankofamatrix) (th:dimofrowA)

Rank-Nullity Theorem for linear transformations (th:ranknullityforT)

Rank-Nullity Theorem for matrices (th:matrixranknullity)

Rayleigh quotients

reduced row echelon form (def:rref)

redundant vectors (def:redundant)

row echelon form (def:ref)

row equivalent matrices

row matrix (vector)

row space of a matrix (def:rowspace)

S

Scalar

scalar triple product

similar matrices (def:similar)

singular matrix (def:nonsingularmatrix)

singular value decomposition (SVD)

singular values (singularvalues)

skew symmetric matrix (def:symmetricandskewsymmetric)

span of a set of vectors (def:span)

spanning set

spectral decomposition - another name for eigenvalue decomposition (def:eigdecomposition)

Spectral Theorem

spectrum The set of distinct eigenvalues of a matrix.

standard basis (def:standardbasis)

standard matrix of a linear transformation (def:standardmatoflintrans)

Standard Position

Standard Unit Vectors (def:standardunitvecrn)

strictly diagonally dominant (def:strict_diag_dom)

square matrix

subspace (def:subspaceabstract)

subspace of (def:subspace)

subspace test (th:subspacetestabstract)

Subtraction of vectors

symmetric matrix (def:symmetricandskewsymmetric)

system of linear equations

T

trace of a matrix (def:trace)

transpose of a matrix (def:matrixtranspose)

triangle inequality

U

Unit Vector

upper triangular matrix

V

Vector

Vector equation of a line (form:vectorlinend)

vector space () (def:vectorspacegeneral)

W

X

Y

Z

zero matrix (def:zeromatrix)

zero transformation (def:zerotransonrn)