# Linear Algebra/Index
## A

accuracy
    of Gauss' method
addition
    vector
additive inverse
adjoint matrix
angle
antipodal
antisymmetric matrix
argument
Arithmetic-Geometric Mean Inequality
arrow diagram 1, 2, 3, 4, 5
augmented matrix
automorphism
    dilation
    reflection
    rotation

## B

back-substitution
base step
    of induction
basis 1, 2, 3
    change of
    definition
    natural
    orthogonal
    orthogonalization
    orthonormal
    standard 1, 2
    standard over the complex numbers
    string
best fit line
binary relation
block matrix
box
    orientation
    sense
    volume

## C

C language
classes
    equivalence
canonical form
    for row equivalence
    for matrix equivalence
    for nilpotent matrices
    for similarity
canonical representative
Cauchy-Schwarz Inequality
Caley-Hamilton theorem
change of basis
characteristic
    equation
    polynomial
    value
    vector
characterized
Chemistry problem 1, 2, 3
central projection
circuits
    parallel
    series
    series-parallel
closure
    of rangespace
    of nullspace
codomain
cofactor
column
    vector
column rank
    full
column space
complement
complementary subspaces
    orthogonal
complex numbers
    vector space over
component
composition
    self
computer algebra systems
concatenation
condition number
congruent figures
congruent plane figures
contradiction
contrapostivite
convex set
coordinates
    homogeneous
    with respect to a basis
corollary
correspondence 1, 2
cosets
Cramer's Rule
cross product
crystals
    diamond
    graphite
    salt
    unit cell

## D

da Vinci, Leonardo
determinant 1, 2
    cofactor
    Cramer's Rule
    definition
    exists 1, 2, 3
    Laplace Expansion
    minor
    Vandermonde
    permutation expansion 1, 2
diagonal matrix 1, 2
diagonalizable
difference equation
    homogeneous
dilation
    matrix representation
dimension
    physical
dilation 1, 2
direct map
direct sum
    definition
    of two subspaces
    external
    internal
direction vector
distance-preserving
division theorem
domain
dot product
double precision
    dual space

## E

echelon form
    leading variable
    free variable
    reduced
eigenvalue
    of a matrix
    of a transformation
eigenvector
    of a matrix
    of a transformation
eigenspace
element
elementary
    matrix
elementary reduction matrices
elementary reduction operations
    pivoting
    rescaling
    swapping
elementary row operations
empty
Erlanger Program
entry
equivalence
    class
       canonical representative
    representitive
equivalence relation 1, 2
    row equivalence
    isomorphism
    matrix equivalence
    matrix similarity
equivalent statements
Euclid
even functions 1, 2
even polynomials
external direct sum

## F

Fibonacci sequence
field
    definition
finite-dimensional vector space
flat
form
free variable
full column rank
full row rank
function 1, 2
    argument
    codomain
    composition
    composition
    correspondence
    domain
    even
    identity
    inverse 1, 2
    inverse image
    left inverse
    multilinear
    range
    restriction
    odd
    one-to-one function
    onto
    right inverse
    structure preserving 1, 2
       see homomorphism
    two sided inverse
    value
    well-defined
    zero
Fundamental Theorem
    of Linear Algebra

## G

Gauss' Method
    accuracy
    back-substitution
    elementary operations
    Gauss-Jordan
Gauss-Jordan
Gaussian operations
generalized nullspace
generalized rangespace
Gram-Schmidt Orthogonalization
Geometry of Eigenvalues
Geometry of Linear Maps

## H

historyless
    Markov Chain
homogeneous
homogeneous coordinate vector
homogeneous coordinates
homorphism
    composition
    matrix representation 1, 2, 3
    nonsingular 1, 2
    nullity
    nullspace
    rank 1, 2
    rangespace
    rank
    zero

## I

ideal line
ideal point
identity function
identity matrix 1, 2
identity function
if-then statement
ill-conditioned
image
    under a function
improper subspace
incidence matrix
index
    of nilpotentcy
induction 1, 2
inductive step
    of induction
inherited operations
inner product
Input-Output Analysis
internal direct sum 1, 2
intersection
invariant subspace
    definintion
inverse
    additive
    left inverse
    matrix
    right inverse
    two-sided
inverse function
inverse image
inversion
isometry
isomorphism 1, 2, 3
    characterized by dimension
    definition
    of a space with itself

## J

Jordon form
    represents similarity classes
Jordon block

## K

kernel
Kirkoff's Laws
Klein, F.

## L

Laplace Expansion
leading variable
least squares
lemma
length
Leontief, W.
line
    at infinity
    in projective plane
    of best fit
linear
    tranpose operation
linear combination
Linear Combination Lemma
linear equation
    coefficients
    constant
    homogeneous
    solution of
       Cramer's Rule
       Gausses' Method
       Gauss-Jordan
    system of
    satisfied by a vector
linear map
    dilation
    see homomorphism
    reflection
    rotation 1, 2
    skew
    trace
linear recurrence
linear relationship
linear surface
linear transformation
linearly dependent
linearly independent
LINPACK

## M

map
    extended linearly
    distance-preserving
    self composition
Maple
Markov Chain
    historyless
Markov matrix
material implication
Mathematica
mathematical induction1, 2
MATLAB
matrix
    adjoint
    antisymmetric
    augmented
    block 1, 2
    change of basis
    characteristic polynomial
    cofactor
    column
    column space
    condition number
    determinant 1, 2
    diagonal matrix 1, 2
    diagonalizable
    diagonalized
    eigenvalue
    eigenvector
    elementary reduction 1, :2
    entry
    equivalent
    form
    identity 1, 2
    incidence
    inverse 1, 2
       existence
    left inverse
    main diagonal
    Markov
    minimal polynomial
    minor
    nilpotent
    nonsingular
    orthogonal
    orthonormal
    right inverse
    scalar multiple
    skew-symmetric
    similar
    similarity
    singular
    submatrix
    sum
    symmetric 1, 2, 3,4, 5, 6
    trace 1, 2, 3
    transition
    transpose 1, 2, 3
    Markov
    matrix-vector product
    minimal polynomial
    multiplication
    nonsingular 1, 2
    permutation
    principle diagonal
    rank
    representation
    row
    row equivalence
    row rank
    row space
    scalar multiple
    singular
    sum
    symmetric 1, 2, 3,4, 5
    trace 1, 2
    transpose 1, 2, 3
    triangular 1, 2, 3
    unit
    Vandermonde
matrix equivalence
    definition
    cononical form
    rank characterization
mean
    arithmetic
    geometric
member
method of powers
minimal polynomial 1, 2
minor
morphism
multilinear
multiplication
    matrix-matrix
    matrix-vector
MuPAD
mutual inclusion 1, 2

## N

natural representative
networks
    Kirkoff's Laws
nilpotent
    canonical form for
    definition
    matrix
    transformation
nilpotentcy
    index
nonsingular
    homomorphism
    matrix
normalize
nullity
nullspace
    closure of
    generalized

## O

Octave
odd functions 1, 2
odd polynomials
one-to-one function
onto function
opposite map
order of a recurrence
ordered pair
orientation 1, 2
orthogonal
    basis
    complement
    mutually
    projection
    matrix
orthogonalization
orthonormal basis
orthonormal matrix

## P

pair
    ordered
parallelepiped
parallelogram rule
parameter
partition
    matrix equivalence classes
partial pivoting
partitions
    row equivalence classes
    isomorphism classes
    matrix equivalence classes
Pascal's triangle
permutation
    inversions
    matrix
    signum
permutation expansion 1, 2
perp
perpendicular
permutation expansion
perspective
    triangles
Physics problem
pivoting
    on rows
    full
    partial
       scaled
plane figure
    congruence
polynomial
    division theorem
    even
    odd
    of a map
    of a matrix
    minimal
point
    at infinity
    in the projective plane
populations, stable
potential
powers, method of
preserves structure
probability vector
projection 1, 2, 3, 4
    along a subspace
    central
       vanishing point
    onto a line
    onto a subspace
    orthogonal 1, 2
Projective Geometry
projective plane
    Duality Principle
    ideal line
    ideal points
    lines
proof techniques
    induction
proper
    subspace
    subset
propositions
    equivalent

## Q

quantifier 1, 2
    existential
    universal

## R

range
rangespace
    closure of
    generalized
rank 1, 2
    column
    of a homomorphism 1, 2
    row
recurrence 1, 2
    homogeneous
    initial conditions
reduced echelon form
reflection
    glide
reflection about a line
    matrix representation
reflexivity
relation
    equivalence
    reflexive
    symmetric
    transitive
relationship
    linear
representation
    of a vector
    of a matrix
representative
    canonical
    for row equivalence
    of matrix equivalence classes
    of similarity classes
rescaling rows
resistance
    equivalent
resistor
restriction
rigid motion
rotation 1, 2, 3
    matrix representation
rounding error
row
    vector
row equivalence
row rank
    full
row space

## S

scaled partial pivoting
scalar
scalar multiple
    matrix
    vector
scalar multiplication
    vector 1, 2
scalar product
Schur's triangularization lemma
Schwarz Inequality
SciLab
self composition
    of maps
sense
sequence
    concatenation
set
    complement
    element
    empty
    empty
    intersection
    linearly dependent
    linearly independent
    member
    mutual inclusion 1, :2
    proper subset
    span of
    subset
    union
signum
similar 1, 2, 3
    canonical form
similar triangles
similarity transformation
single precision
singular
    matrix
size 1, 2
sgn
    see signum
skew
skew-symmetric
span
    of a singleton
spin
square root
stable populations
standard basis
state
    absorbing
Statics problem
string
    basis
    of basis vectors
structure
    preservation
submatrix
subset
subspace
    closed
    complementary
    direct sum
    definition
    improper
    independence
    invariant
    orthogonal complement
    proper
    sum
sum
    of matrices
    of subspaces
    vector 1, 2, 3, 4
summation notation
    for permutation expansion
swapping rows
symmetric matrix 1, 2, 3, 4
symmetry
system of linear equations
    Gauss' Method
    solving

## T

theorem
trace 1, 2, 3
transformation
    characteristic polynomial
    composed with itself
    diagonalizable
    eigenvalue
    eigenvector
    eigenspace
    Jordon form
    minimal polynomial
    nilpotent
       canonical representative
    projection
    size change
transition matrix
transitivity
translation
transpose 1, 2
    interaction with sum and scalar multiplication
    determinant 1, 2
triangles
    similar
Triangle Inequality
triangular matrix
triangularization
trivial space 1, 2
turning map
    matrix representation

## U

union
unit matrix

## V

vacuously true
value
Vandermonde
    determinant
    matrix
vanishing point
vector 1, 2
    angle between vecots
    canonical position
    closure
    column
    complex scalars
    component
    cross product
    direction
    dot product
    free
    homogeneous coordinate
    length
    natural position
    orthogonal
    probability
    representation of 1, 2
    row
    satisfies an equation
    scalar multiple
    scalar multiplication 1, 2
    sum 1, 2, 3, 4
    unit
    zero vector 1, 2
vector space
    basis
    definition 1, 2
    dimension
    dual
    finite-dimensional
    homomorphism
    isomorphism
    map
    over complex numbers
    subspace
    trivial 1, 2
Venn diagram
voltage drop
volume
voting paradox:
    majority cycle
    rational preference
    spin
voting paradoxes

## W

Wheatstone bridge
well-defined

## Z

zero divisor 1, 2
zero homomorphism
zero vector 1, 2