INDEX
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index.
A
addition of vectors
adjugate of a matrix
algebraic multiplicity of an eigenvalue
associated homogeneous system (def:asshomsys )
Given any linear system , the system is called the associated homogeneous system .
augmented matrix
B
back substitution
basis (def:basis )
A set of vectors is called a basis of (or a basis of a subspace of ) provided
that
(a) (or )
(b) is linearly independent.
block matrices
box product
C
change-of-basis matrix (def:matlintransgenera )
characteristic equation (def:chareqcharpoly )
The equation
is called the characteristic equation of .
characteristic polynomial
Cholesky factorization
closed under addition (def:closedunderaddition )
A set is said to be closed under addition if for each element and the sum is also in .
closed under scalar multiplication (def:closedunderscalarmult )
A set is said to be closed under scalar multiplication if for each element and for each
scalar the product is also in .
codomain of a linear transformation
coefficient matrix
cofactor expansion
column matrix (vector)
column space of a matrix (def:colspace )
Let be an matrix. The column space of , denoted by , is the subspace of spanned by
the columns of .
composition of linear transformations (def:compoflintrans )
Let , and be vector spaces, and let and be linear transformations. The composition
of and is the transformation given by
consistent system
convergence
coordinate vector with respect to a basis (in ) (abstract vector spaces )
Cramer’s Rule
cross product (def:crossproduct )
Let and be vectors in . The cross product of and , denoted by , is given
by
D
determinant () (def:twodetcrossprod )
A determinant is a number associated with a matrix
determinant () (def:threedetcrossprod )
A determinant is a number associated with a matrix
diagonal matrix
diagonalizable matrix (def:diagonalizable )
Let be an matrix. Then is said to be diagonalizable if there exists an invertible
matrix such that
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 )
Let be a subspace of . The dimension of is the number, , of elements in any basis of .
We write
Direction vector
Distance between points in (form:distRn )
Let and be points in . The distance between and is given by
Distance between point and line
divergence
domain of a linear transformation
dominant eigenvalue (def:dominant ew,ev )
An eigenvalue of an matrix is called a dominant eigenvalue if has multiplicity , and
Any corresponding eigenvector is called a
dominant eigenvector of .
dot product (def:dotproduct )
Let and be vectors in . The dot product of and , denoted by , is given
by
E
eigenspace (def:eigspace )
The set of all eigenvectors associated with a given eigenvalue of a matrix is known as
the eigenspace associated with that eigenvalue.
eigenvalue (def:eigen )
Let be an matrix. We say that a non-zero vector is an eigenvector of
if
for some scalar . We say that is an eigenvalue of associated with the eigenvector .
eigenvalue decomposition (def:eigdecomposition )
If we are able to diagonalize , say , we say that is an eigenvalue decomposition of .
eigenvector (def:eigen )
Let be an matrix. We say that a non-zero vector is an eigenvector of
if
for some scalar . We say that is an eigenvalue of associated with the eigenvector .
elementary matrix (def:elemmatrix )
An elementary matrix is a square matrix formed by applying a single elementary row
operation to the identity matrix.
elementary row operations (def:elemrowops )
The following three operations performed on a linear system are called elementary
row operations .
(a) Switching the order of equations (rows) and :
(b) Multiplying both sides of equation (row) by the same non-zero constant,
, and replacing equation with the result:
(c) Adding times equation (row) to equation (row) , and replacing equation
with the result:
equivalence relation
equivalent linear systems (def:equivsystems )
Two systems of linear equations are said to be equivalent if they have the same
solution set.
F
free variable
fundamental subspaces of a matrix
G
Gaussian elimination (def:GaussianElimination )
The process of using the elementary row operations on a matrix to transform it into
row-echelon form is called Gaussian Elimination .
Gauss-Jordan elimination (def:GaussJordanElimination )
The process of using the elementary row operations on a matrix to transform it into
reduced row-echelon form is called Gauss-Jordan elimination .
Gauss-Seidel method
geometric multiplicity of an eigenvalue (def:geommulteig )
The geometric multiplicity of an eigenvalue is the dimension of the corresponding
eigenspace .
Gerschgorin’s Disk Theorem
Gram-Schmidt Process
H
homogeneous system (def:homogeneous )
A system of linear equations is called homogeneous if the system can be written in
the form
hyperplane
I
identity matrix
identity transformation (def:idtransonrn )
The identity transformation on , denoted by , is a transformation that maps each
element of to itself.
In other words,
is a transformation such that
image of a linear transformation (def:imageofT )
Let and be vector spaces, and let be a linear transformation. The image of ,
denoted by , is the set
In other words, the image of consists of individual images of all vectors of .
inconsistent system
inner product (def:innerproductspace )
An inner product on a real vector space is a function that assigns a real number to
every pair , of vectors in in such a way that the following properties are
satisfied.
(a) is a real number for all and in .
(b) for all and in .
(c) for all , , and in .
(d) for all and in and all in .
(e) for all in .
inner product space
inverse of a linear transformation (def:inverseoflintrans )
Let and be vector spaces, and let be a linear transformation. A transformation
that satisfies and is called an inverse of . If has an inverse, is called invertible .
inverse of a square matrix (def:matinverse )
Let be an matrix. An matrix is called an inverse of if
where is an identity matrix. If such an inverse matrix exists, we say that
is invertible . If an inverse does not exist, we say that is not invertible.
isomorphism (def:isomorphism )
Let and be vector spaces. If there exists an invertible linear transformation we say
that and are isomorphic and write . The invertible linear transformation is called
an isomorphism .
iterative methods
J
Jacobi’s method
K
kernel of a linear transformation (def:kernel )
Let and be vector spaces, and let be a linear transformation. The kernel of ,
denoted by , is the set
In other words, the kernel of consists of all vectors of that map to in .
L
Laplace Expansion Theorem (th:laplace1 )
leading entry (leading 1) (def:leadentry )
The first non-zero entry in a row of a matrix (when read from left to right) is called
the leading entry . When the leading entry is 1, we refer to it as a leading 1 .
leading variable
linear combination of vectors (def:lincomb )
A vector is said to be a linear combination of vectors if
for some scalars .
linear equation (def:lineq )
A linear equation in variables is an equation that can be written in the
form
where and are constants.
linear transformation (def:lin ) (also see Linear Transformations of Abstract Vector Spaces )
A transformation is called a linear transformation if the following are true for all
vectors and in , and scalars .
linearly dependent vectors (def:linearindependence )
Let be vectors of . We say that the set is linearly independent if the only solution to
is the
trivial solution .
If, in addition to the trivial solution, a non-trivial solution (not all are zero) exists,
then we say that the set is linearly dependent .
linearly independent vectors (def:linearindependence )
Let be vectors of . We say that the set is linearly independent if the only solution to
is the
trivial solution .
If, in addition to the trivial solution, a non-trivial solution (not all are zero) exists,
then we say that the set is linearly dependent .
lower triangular matrix
LU factorization
M
Magnitude of a vector (def:normrn )
Let be a vector in , then the length , or the magnitude , of is given by
main diagonal
matrix
matrix addition (def:additionofmatrices )
Let and be two matrices. Then the sum of matrices and , denoted by , is an
matrix given by
matrix equality (def:equalityofmatrices )
Let and be two matrices. Then means that for all and .
matrix multiplication (by a matrix) (def:matmatproduct )
Let be an matrix whose rows are vectors , . Let be an matrix with columns . Then
the entries of the matrix product are given by the dot products
matrix multiplication (by a scalar) (def:scalarmultofmatrices )
If and is a scalar, then .
matrix multiplication (by a vector) (def:matrixvectormult )
Let be an matrix, and let be an vector. The product is the vector given
by:
or, equivalently,
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 )
A square matrix is said to be nonsingular provided that . Otherwise we say that is
singular .
normal vector
null space of a matrix (def:nullspace )
Let be an matrix. The null space of , denoted by , is the set of all vectors in such
that .
nullity of a linear transformation (def:nullityT )
The nullity of a linear transformation , is the dimension of the kernel of
.
nullity of a matrix (def:matrixnullity )
Let be a matrix. The dimension of the null space of is called the nullity of
.
O
one-to-one (def:onetoone )
A linear transformation is one-to-one if
onto (def:onto )
A linear transformation is onto if for every element of , there exists an element of
such that .
ordered basis
orthogonal basis
orthogonal complement of a subspace of (def:023776 )
If is a subspace of , define the orthogonal complement of (pronounced “-perp”) by
Orthogonal Decomposition Theorem (th:OrthoDecomp )
Let be a subspace of and let . Then there exist unique vectors and such that .
orthogonal matrix (def:orthogonal matrices )
An matrix is called an orthogonal matrix if it satisfies one (and hence all) of the
conditions in Theorem th:orthogonal_matrices .
Orthogonal projection onto a subspace of (def:projOntoSubspace )
Let be a subspace of with orthogonal basis . If is in , the vector
is called the
orthogonal projection of onto .
Orthogonal projection onto a vector (def:projection )
Let be a vector, and let be a non-zero vector. The projection of onto is given
by
orthogonal set of vectors (orthset )
Let be a set of nonzero vectors in . Then this set is called an orthogonal set if for all
. Moreover, if for (i.e. each vector in the set is a unit vector), we say the set of
vectors is an orthonormal set .
Orthogonal vectors (def:orthovectors )
Let and be vectors in . We say and are orthogonal if .
orthogonally diagonalizable matrix (def:orthDiag )
An matrix is said to be orthogonally diagonalizable if an orthogonal matrix can be
found such that is diagonal.
orthonormal basis
orthonormal set of vectors (orthset )
Let be a set of nonzero vectors in . Then this set is called an orthogonal set if for all
. Moreover, if for (i.e. each vector in the set is a unit vector), we say the set of
vectors is an orthonormal set .
P
parametric equation of a line (form:paramlinend )
Let be a direction vector for line in , and let be an arbitrary point on . Then the
following parametric equations describe :
particular solution
partitioned matrices (block multiplication)
permutation matrix
pivot
positive definite matrix (def:024811 )
A square matrix is called positive definite if it is symmetric and all its eigenvalues
are positive. We write when eigenvalues are real and positive.
power method (and its variants)
Q
QR factorization (def:QR-factorization )
Let be an matrix with independent columns. A QR-factorization of expresses it as
where is with orthonormal columns and is an invertible and upper triangular
matrix with positive diagonal entries.
R
rank of a linear transformation (def:rankofT )
The rank of a linear transformation , is the dimension of the image of .
rank of a matrix (def:rankofamatrix ) (th:dimofrowA )
The rank of matrix , denoted by , is the number of nonzero rows that remain after we
reduce to row-echelon form by elementary row operations.
For any matrix ,
Rank-Nullity Theorem for linear transformations (th:ranknullityforT )
Let be a linear transformation. Suppose , then
Rank-Nullity Theorem for matrices (th:matrixranknullity )
Rayleigh quotients
reduced row echelon form (def:rref )
A matrix that is already in row-echelon form is said to be in reduced row-echelon
form if:
(a) Each leading entry is
(b) All entries above and below each leading are
redundant vectors (def:redundant )
Let be a set of vectors in . If we can remove one vector without changing the span of
this set, then that vector is redundant . In other words, if
we say that is a redundant element of , or simply redundant.
row echelon form (def:ref )
A matrix is said to be in row-echelon form if:
(a) All entries below each leading entry are 0.
(b) Each leading entry is in a column to the right of the leading entries in the
rows above it.
(c) All rows of zeros, if there are any, are located below non-zero rows.
row equivalent matrices
row matrix (vector)
row space of a matrix (def:rowspace )
Let be an matrix. The row space of , denoted by , is the subspace of spanned by
the rows of .
S
Scalar
scalar triple product
similar matrices (def:similar )
If and are matrices, we say that and are similar , if for some invertible matrix . In
this case we write .
singular matrix (def:nonsingularmatrix )
A square matrix is said to be nonsingular provided that . Otherwise we say that is
singular .
singular value decomposition (SVD)
singular values (singularvalues )
Let be an matrix. The singular values of are the square roots of the positive
eigenvalues of
skew symmetric matrix (def:symmetricandskewsymmetric )
An matrix is said to be symmetric if It is said to be skew symmetric if
span of a set of vectors (def:span )
Let be vectors in . The set of all linear combinations of is called the span of . We
write
and we say that vectors span . Any vector in is said to be in the span of . The set
is called a spanning set for .
spanning set
spectral decomposition - another name for eigenvalue decomposition (def:eigdecomposition )
If we are able to diagonalize , say , we say that is an eigenvalue decomposition of .
Spectral Theorem
spectrum The set of distinct eigenvalues of a matrix.
standard basis (def:standardbasis )
The set is called the standard basis of .
standard matrix of a linear transformation (def:standardmatoflintrans )
The matrix in Theorem th:matlin is known as the standard matrix of the linear transformation
.
Standard Position
Standard Unit Vectors (def:standardunitvecrn )
Let denote a vector that has as the component and zeros elsewhere. In other
words,
where is in the position. We say that is a standard unit vector of .
strictly diagonally dominant (def:strict_diag_dom )
Let be the matrix which is the coefficient matrix of the linear system .
Let
denote the sum of the absolute values of the non-diagonal entries in row . We say
that is strictly diagonally dominant if
for all values of from to .
square matrix
subspace (def:subspaceabstract )
A nonempty subset of a vector space is called a subspace of , provided that is itself
a vector space when given the same addition and scalar multiplication as .
subspace of (def:subspace )
Suppose that is a nonempty subset of that is closed under addition and closed under
scalar multiplication. Then is a subspace of .
subspace test (th:subspacetestabstract )
Let be a nonempty subset of a vector space . If is closed under the operations of
addition and scalar multiplication of , then is a subspace of .
Subtraction of vectors
symmetric matrix (def:symmetricandskewsymmetric )
An matrix is said to be symmetric if It is said to be skew symmetric if
system of linear equations
T
trace of a matrix (def:trace )
The trace of an matrix , abbreviated , is defined to be the sum of the main diagonal
elements of . In other words, if , then
We may also write .
transpose of a matrix (def:matrixtranspose )
Let be an matrix. Then the transpose of , denoted by , is the matrix given by
triangle inequality
U
Unit Vector
upper triangular matrix
V
Vector
Vector equation of a line (form:vectorlinend )
Let be a direction vector for line in , and let be an arbitrary point on . Then the
following vector equation describes :
vector space ( ) (def:vectorspacegeneral )
Let be a nonempty set. Suppose that elements of can be added together and
multiplied by scalars. The set , together with operations of addition and scalar
multiplication, is called a vector space provided that
is closed under addition
is closed under scalar multiplication
and the following properties hold for , and in and scalars and :
(a) Commutative Property of Addition:
(b) Associative Property of Addition:
(c) Existence of Additive Identity:
(d) Existence of Additive Inverse:
(e) Distributive Property over Vector Addition:
(f) Distributive Property over Scalar Addition:
(g) Associative Property for Scalar Multiplication:
(h) Multiplication by :
We will refer to elements of as vectors .
W
X
Y
Z
zero matrix (def:zeromatrix )
The zero matrix is the matrix having every entry equal to zero. The zero matrix is
denoted by .
zero transformation (def:zerotransonrn )
The zero transformation , , maps every element of the domain to the zero
vector.
In other words,
is a transformation such that