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Index Index

addition
matrix, Item
vector, Item
adjoint
classical, Discovery Item
matrix (relative to an inner product), Item
algebraic multiplicity, Discovery Item
argument (of a complex number), Paragraph
augmented matrix, Discovery Item
axis
imaginary, Paragraph
real, Paragraph
basis, Item
matrix, change of, Item
ordered, Item
standard, Discovery Item
transition matrix, Item
best approximation, Item
block
Jordan, Item
of a matrix, Item
block (in a matrix), Item
block-diagonal form, Item
Cauchy-Schwarz inequality, Theorem Theorem Theorem
Cayley-Hamilton Theorem, Theorem
change of basis matrix, Item
characteristic
equation (for a matrix), Item
equation (for an operator), Item
polynomial, Discovery
polynomial (for a matrix), Item
polynomial (for an operator), Item
space, Note
value, Note
vector, Note
codomain (of a linear transformation), Item
coefficient matrix, Item
cofactor, Discovery Item
expansion, Item
matrix, Discovery Item
column
in a matrix, Item
space, Discovery Item
vector, Item
companion
matrix, Item
complement
orthogonal, Item
subspace, Proposition
complex
argument, Paragraph
conjugate, Definition
inner product, Item Definition
inner product space, Item
linear equation, Item
matrix, Item
adjoint, Item
conjugate, Item
Hermitian, Item
self-adjoint, Item
modulus, Definition Definition
number
polar form, Paragraph
numbers, Definition
imaginary part, Definition
real part, Definition
pairing, Remark
plane, Subsection
polynomial, Paragraph
unit circle, Paragraph
vector
\(n\)-dimensional, Item
component
orthogonal, Discovery
orthogonal (to a subspace of an inner product space), Item
orthogonal (to a vector in \(\R^n\)), Item
parallel, Discovery
parallel (to a subspace of an inner product space), Item
parallel (to a vector in \(\R^n\)), Item
components, Discovery Item
composite, Item
consistent system, Item
coordinate
transformation, Item
coordinate vector, Discovery Item Paragraph
coordinates, Discovery Item Paragraph
Cramer's rule, Subsection
cross product, Item Paragraph
cross terms, Item
cyclic
space, Item
degree of nilpotency, Item
DeMoivre's formula, Corollary
determinant, Paragraph Discovery
linear operator, Item
matrix, Item
minor, Discovery Item
diagonal matrix, Item Item
diagonalizable, Item
orthogonally, Item
unitarily, Item
dimension, Item
directed line segment, Discovery Item
distance
between a vector and a subspace, Item
domain (of a linear transformation), Item
dot product, Discovery Item
complex, Item
dual
basis, Theorem
double, Paragraphs
space, Item
eigenspace, Discovery
generalized, Item
matrix, Item
operator, Item
eigenvalue
matrix, Item
multiplicity
algebraic, Discovery Item
geometric, Discovery Item
operator, Item
eigenvector, Discovery
generalized, Item
matrix, Item
operator, Item
elementary
matrix, Paragraph Item
nilpotent form, Item
row operations, Item
entry (of a matrix), Item
equation(s)
linear, Item
complex, Item
parametric, Item
system of, Item
Euclidean
inner product, Discovery Item
Euler's formula, Paragraph
field
of abstract numbers, Paragraph
of complex numbers, Definition
finite-dimensional, Item Item
form
block-diagonal, Item
elementary nilpotent, Item
Jordan normal form, Item
scalar-triangular, Item
triangular block, Item
triangular-block nilpotent, Item
free variables, Item
functional, Item
Fundamental Theorem of Algebra
complex version, Theorem
real version, Theorem
general solution, Item
generalized
eigenspace, Item
eigensubspace, Item
eigenvector, Item
geometric multiplicity, Discovery Item
homogeneous system, Discovery Item
homomorphism (of vector spaces), Item
identity
operator, Item
identity matrix, Item
image, Item
imaginary
axis, Paragraph
number, Definition
purely, Remark
part of a complex number, Definition
inconsistent system, Item
independent
subspaces, Item
complete set, Item
inequality
Cauchy-Schwarz, Theorem Theorem Theorem
triangle, Theorem Theorem
initial point, Discovery Item
injective, Item
inner product
complex, Item Definition
Euclidean, Discovery Discovery Item
real, Item Definition
space (complex), Item
space (real), Item
standard (on \(\C^n\)), Item
standard (on \(\R^n\)), Item
invariant subspace, Item
inverse
pseudo-, Item
inverse (of a matrix), Item
invertible, Discovery
matrix, Item
isomorphic, Item
isomorphism, Item
Jordan
block, Item
normal form, Item
kernel, Item
leading
one, Item
variables, Item
least-squares solution, Item
level set, Item
linear
combination, Paragraph Discovery Item
of vectors, Item
equation, Item
complex, Item
functional, Item
operator, Item
identity, Item
scalar, Item
zero, Item
transformation, Item
coordinate, Item
inverse, Item
invertible, Item
space, Item
zero, Item
linearly
dependent, Discovery Item
independent, Discovery Item
lower triangular matrix, Item Item
main diagonal, Item Paragraph
matix
diagonal, Item Item
lower triangular, Item Item
symmetric, Item Item
upper triangular, Item Item
matrix, Item
addition, Item
adjoint (classical), Discovery Item
adjoint (relative to an inner product), Item
augmented, Discovery Item
block-diagonal, Item
change of basis, Item
characteristic equation, Item
characteristic polynomial, Item
coefficient, Item
cofactor, Discovery Item
cofactor expansion, Item
column, Item
column space, Discovery Item
companion, Item
complex, Item
adjoint, Item
conjugate, Item
determinant, Discovery Item
diagonalizable, Item
orthogonally, Item
unitarily, Item
dimensions, Discovery Item
eigenspace, Item
eigenvalue, Item
eigenvector, Item
elementary, Item
elementary nilpotent, Item
entry, Item Item
equality, Item
Hermitian, Item
identity, Item
inverse, Item
invertible, Item
minor, Discovery Item
nilpotent, Paragraph
normal (relative to an inner product), Item
null space, Paragraph Item
nullity, Item
of cofactors, Discovery Item
orthogonal, Item
positive definite (complex), Item
positive definite (real), Item
product-preserving, Item
row, Item
row space, Paragraph Item
scalar, Item Item
scalar multiple, Item
scalar-triangular, Item
self-adjoint, Item
self-adjoint (relative to an inner product), Item
similarity, Item Item
singular, Item
size, Discovery Item
square, Item
standard (of a linear transformation \(\R^n \to \R^m\)), Item
sub-, Item
transformation (complex), Item
transformation (real), Item
transition, Item Discovery Item
transpose, Item
triangular block, Item
triangular-block nilpotent, Item
unipotent, Paragraph
unitarily diagonalizable, Item Item
unitary, Item
Vandermonde, Item
zero, Item
minor, Discovery Item
modulus (of a complex number), Definition
multiplicity
algebraic, Discovery Item
geometric, Discovery Item
nilpotent
matrix, Paragraph Item
triangular-block form, Item
nontrivial solution, Item
norm, Discovery
in \(\R^n\), Item
in an inner product space, Item
normal
matrix (relative to an inner product), Item
system, Item
normal vector, Discovery Item
normalization, Item
null
space, Paragraph Item
nullity
of a linear transformation, Item
of a matrix, Item
one-to-one, Item
onto, Item
operator, Item
characteristic equation, Item
characteristic polynomial, Item
determinant, Item
eigenspace, Item
eigenvalue, Item
eigenvector, Item
linear
identity, Item
scalar, Item
zero, Item
trace, Item
ordered basis, Item
orthogonal
complement, Item
component, Discovery
component (to a subspace of an inner product space), Item
component (to a vector in \(\R^n\)), Item
matrix, Item
projection, Discovery
projection (onto a subspace in an inner product space), Item
projection (onto a vector in \(\R^n\)), Item
set, Item
vectors, Discovery
vectors (in \(\R^n\)), Item
vectors (in an inner product space), Item
orthogonally diagonalizable matrix, Item
orthonormal set, Item
pairing, Item
complex, Remark
real, Remark
parallel
component, Discovery
component (to a subspace of an inner product space), Item
component (to a vector in \(\R^n\)), Item
vectors, Item
parallel vectors, Paragraph
parallelogram rule, Paragraph
parametric equations, Item
point-normal form
for a line, Discovery Item Paragraph
for a plane, Item
point-parallel form
for a line, Item
for a plane, Item
polar form (of a complex number), Paragraph
positive definite matrix
complex, Item
real, Item
principal axes, Item
product-preserving
matrix, Item
projection, Discovery
onto a subspace in an inner product space, Item
onto a vector in \(\R^n\), Item
pseudo-inverse matrix, Item
quadratic form, Item
diagonal, Item
range, Item
rank
of a linear transformation, Item
of a matrix, Item
real
axis, Paragraph
inner product, Item Definition
inner product space, Item
pairing, Remark
part of a complex number, Definition
right-hand rule, Paragraph
root
of unity, Paragraph
row
echelon form, Item
reduced, Item
equivalent, Item
in a matrix, Item
reduction, Item
space, Paragraph Item
vector, Item
row operations
elementary, Item
scalar, Item Item Item
matrix, Item Item
multiple
of a matrix, Item
of a vector, Item
multiplication
abstract (complex), Item
abstract (real), Item
operator, Item
scalar-triangular form, Item
self-adjoint
matrix (relative to an inner product), Item
similar matrices, Item Item
similarity class, Item
singular, Discovery
matrix, Item
size (of a matrix), Item
solution
general, Item
least-squares, Item
nontrivial, Item
set, Item
space, Item
to a system of linear equations, Item
trivial, Item
span, Discovery Item
spanning
set, Item
square matrix, Item
standard
basis, Discovery
basis vectors, Item
inner product, Discovery
inner product (on \(\C^n\)), Item
inner product (on \(\R^n\)), Item
matrix (of a linear transformation \(\R^n \to \R^m\)), Item
submatrix, Item
subspace, Item
generated, Item
independent, Item
complete set, Item
invariant, Item
test, Paragraphs Procedure Proposition
Subspace Test, Paragraphs Procedure Proposition
surjective, Item
symmetric matrix, Item Item
system
consistent, Item
homogeneous, Discovery Item
inconsistent, Item
normal, Item
of differential equations, Item Item
of linear equations, Item
solution, Item
terminal point, Discovery Item
Theorem
Cayley-Hamilton, Theorem
trace
linear operator, Item
transformation
inverse, Item
invertible, Item
linear, Item
coordinate, Item
space, Item
zero, Item
matrix (complex), Item
matrix (real), Item
transition matrix, Item Discovery Item
transpose, Item
triangle inequality, Theorem Theorem
triangular block form, Item
triangular-block nilpotent form, Item
trivial
solution, Item
subspace, Item
vector space, Item
unipotent matrix, Paragraph
unit
circle
complex, Paragraph
unit vector, Discovery
in \(\R^n\), Item
in an inner product space, Item
unitarily diagonalizable matrix, Item
unitary
matrix, Item
upper triangular matrix, Item Item
Vandermonde matrix, Item
vector, Discovery Item
abstract, Item Definition
addition, Item Item
angle (in \(\R^n\)), Item
angle (in a real inner product space), Item
column, Item
components, Discovery Item
coordinate, Discovery Item Paragraph
coordinates, Discovery Item Paragraph
cross product, Item Paragraph
distance, Item
initial point, Discovery Item
length, Item
linear combination, Item
magnitude, Item
negative, Item Item
norm, Discovery
norm (in \(\R^n\)), Item
norm (in an inner product space), Item
normal, Discovery Item
normalization, Item
of constants, Item
of unknowns, Item
orthogonal, Discovery
orthogonal (in \(\R^n\)), Item
orthogonal (in an inner product space), Item
parallel, Item Paragraph
row, Item
scalar multiple, Item
space, Definition
basis, Item
dimension, Item
finite-dimensional, Item
infinite-dimensional, Item
trivial, Item
space (complex), Item
space (real), Item
space homomorphism, Item
standard basis, Discovery Item
subtraction, Item Item
terminal point, Discovery Item
unit, Discovery
unit (in \(\R^n\)), Item
unit (in an inner product space), Item
zero, Item
zero
matrix, Item
operator, Item
transformation, Item
vector, Item Item
space, Item