# Linear Algebra/Index

Contents: Top - 0–9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

## A

accuracy

of Gauss' method

vector

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

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

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

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

kernel

Kirkoff's Laws

Klein, F.

## L

Laplace Expansion

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

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

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

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

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

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

majority cycle
rational preference
spin

## W

Wheatstone bridge

well-defined

## Z

zero divisor 1, 2

zero homomorphism

zero vector 1, 2