FCLA Index

Index Index

additive associativity

column vectors, Property

complex numbers, Property

matrices, Property

vectors, Property

additive closure

column vectors, Property

complex numbers, Property

matrices, Property

vectors, Property

additive commutativity

complex numbers, Property

additive inverse

complex numbers, Property

from scalar multiplication, Theorem

additive inverses

column vectors, Property

matrices, Property

unique, Theorem

vectors, Property

adjoint, Definition

inner product, Theorem

notation, Definition

of a matrix sum, Theorem

of an adjoint, Theorem

of matrix scalar multiplication, Theorem

Archetype A

column space, Example

linearly dependent columns, Example

singular matrix, Example

solving homogeneous system, Example

system as linear combination, Example

archetype A

augmented matrix, Example

Archetype B

column space, Example

inverse, Example

linearly independent columns, Example

nonsingular matrix, Example

not invertible, Example

solutions via inverse, Example

solving homogeneous system, Example

system as linear combination, Example

vector equality, Example

archetype B

solutions, Example

Archetype C

homogeneous system, Example

Archetype D

column space, original columns, Example

solving homogeneous system, Example

vector form of solutions, Example

Archetype I

casting out vectors, Example

column space from row operations, Example

null space, Example

row space, Example

vector form of solutions, Example

Archetype L

null space span, linearly independent, Example

vector form of solutions, Example

augmented matrix, Sage

bases, Sage

basis, Definition

columns nonsingular matrix, Example

common size, Theorem

crazy vector space, Example

matrices, Example Example

polynomials, Example Example Example Example

subspace of matrices, Example

best cities

money magazine, Example

change of basis

between polynomials, Example

change-of-basis, Theorem

between column vectors, Example

matrix representation, Theorem

similarity, Theorem

change-of-basis matrix, Definition Sage

inverse, Theorem

characteristic polynomial, Definition

degree, Theorem

size 3 matrix, Example

coefficient matrix, Definition

nonsingular, Theorem

coercion, Sage

column space

Archetype A, Example

Archetype B, Example

as null space, Example Theorem

as null space, Archetype G, Example

as row space, Theorem

basis, Theorem

consistent system, Theorem

consistent systems, Example Sage

isomorphic to range, Theorem

nonsingular matrix, Theorem

notation, Definition

original columns, Sage

original columns, Archetype D, Example

row operations, Archetype I, Example

subspace, Theorem

testing membership, Example

two computations, Example

column vector addition

notation, Definition

column vector scalar multiplication

notation, Definition

commutativity

column vectors, Property

matrices, Property

vectors, Property

complex \(m\)-space, Example

complex arithmetic, Example

complex number

conjugate, Definition Example

modulus, Definition Example

complex numbers

addition, Definition

notation, Definition

arithmetic properties, Theorem

equality, Definition

notation, Definition

multiplication, Definition

notation, Definition

zero product, Theorem Theorem

complex vector space

dimension, Theorem

composition

injective linear transformations, Theorem

surjective linear transformations, Theorem

conjugate

addition, Theorem

column vector, Definition

matrix, Definition

notation, Definition

multiplication, Theorem

notation, Definition

of conjugate of a matrix, Theorem

scalar multiplication, Theorem

twice, Theorem

vector addition, Theorem

conjugate of a vector

notation, Definition

conjugates, norms, inner products, Sage

conjugation

matrix addition, Theorem

matrix scalar multiplication, Theorem

matrix transpose, Theorem

consistent linear system, Theorem

consistent linear systems, Sage Theorem

consistent system, Definition

consistent systems, spans, Sage

coordinates, Sage

orthonormal basis, Theorem

coordinatization

linear combination of matrices, Example

linear independence, Theorem

orthonormal basis, Example Example

spanning sets, Theorem

coordinatizing

polynomials, Example

crazy vector space, Example

properties, Example

determinant, Definition Sage

bars

notation, Definition

computed two ways, Example

equal rows or columns, Theorem

expansion, columns, Theorem

expansion, rows, Theorem

functional

notation, Definition

identity matrix, Theorem

matrix multiplication, Theorem

nonsingular matrix, Theorem

row or column multiple, Theorem

row or column swap, Theorem

size 2 matrix, Theorem

size 3 matrix, Example

transpose, Theorem

via row operations, Example

zero, Theorem

zero row or column, Theorem

zero versus nonzero, Example

determinant, upper triangular matrix, Example

determinants

elementary matrices, Theorem

diagonal matrix, Definition

diagonalizable, Definition

distinct eigenvalues, Theorem Example

full eigenspaces, Theorem

not, Example

diagonalizable matrix

high power, Example

diagonalization, Example

Archetype B, Example

criteria, Theorem

diagonalization of a matrix, Sage

dimension, Definition Sage

crazy vector space, Example

notation, Definition

polynomial subspace, Example

proper subspaces, Theorem

subspace, Example

dimensions

matrix subspaces, Sage

distributivity

complex numbers, Property

distributivity, matrix addition

matrices, Property

distributivity, scalar addition

column vectors, Property

matrices, Property

vectors, Property

distributivity, vector addition

column vectors, Property

vectors, Property

eigenspace, Definition

as null space, Theorem

subspace, Theorem

eigenvalue, Definition

algebraic multiplicity, Definition

notation, Definition

complex, Example

existence, Theorem Example

geometric multiplicity, Definition

notation, Definition

linear transformation, Definition

multiplicities, Example

power, Theorem

root of characteristic polynomial, Theorem

scalar multiple, Theorem

singular matrix, Theorem

symmetric matrix, Example

zero, Theorem

eigenvalues

building desired, Example

complex, of a linear transformation, Example

computing, Sage

conjugate pairs, Theorem

distinct, Example

Hermitian matrices, Theorem

inverse, Theorem

maximum number, Theorem

multiplicities, Example Theorem

number, Theorem

of a polynomial, Theorem

size 3 matrix, Example Example

transpose, Theorem

eigenvalues and eigenvectors, Sage

eigenvalues, eigenvectors

vector, matrix representations, Theorem

eigenvalues}\index{eigenvectors, Example

eigenvectors

computing, Sage

conjugate pairs, Theorem

Hermitian matrices, Theorem

linear transformation, Example Example

linearly independent, Theorem

of a linear transformation, Example

elementary matrices, Definition Sage

determinants, Theorem

nonsingular, Theorem

row operations, Example Theorem

elementary matrix

add

notation, Definition

multiply

notation, Definition

swap

notation, Definition

empty set

notation, Definition

endomorphisms, Sage

equal matrices

via equal matrix-vector products, Theorem

equation operations, Definition Theorem

equivalent systems, Definition

exact vs.\ inexact computations, Sage

extended echelon form, Sage

submatrices, Example

extended reduced row-echelon form

properties, Theorem

Fibonacci sequence, Example

four subsets, Example Example

four subspaces

dimension, Theorem

free variables, Example

free variables, number, Theorem

free, dependent variables, Sage

free, independent variables, Example

goldilocks, Theorem

Gram-Schmidt

column vectors, Theorem

three vectors, Example

help, Sage

hermitian, Definition

Hermitian matrix

inner product, Theorem

homogeneous system, Definition

Archetype C, Example

consistent, Theorem

infinitely many solutions, Theorem

homogeneous systems

linear independence, Theorem

solving, Sage

identity matrix, Example Sage

determinant, Theorem

notation, Definition

independent, dependent variables, Definition

indesxstring, Example

indexstring, Theorem Theorem Theorem

infinite solution set, Example

infinite solutions, \(3\times 4\), Example

injective, Example Example

not, Example Example Example

not, by dimension, Example

polynomials to matrices, Example

injective linear transformation

bases, Theorem

injective linear transformations

dimension, Theorem

inner product, Example

anti-commutative, Theorem

norm, Theorem

notation, Definition

positive, Theorem

scalar multiplication, Theorem

vector addition, Theorem

inverse, Example Example

composition of linear transformations, Theorem

notation, Definition

invertible linear transformation

defined by invertible matrix, Theorem

invertible linear transformations

composition, Theorem

computing, Example

isomorphic

multiple vector spaces, Example

vector spaces, Example

isomorphic vector spaces, Example

dimension, Theorem

kernel

injective linear transformation, Theorem

isomorphic to null space, Theorem

linear transformation, Example

notation, Definition

of a linear transformation, Definition

preimage, Theorem

subspace, Theorem

trivial, Example

via matrix representation, Example

left null space, Definition Example Sage

as row space, Theorem

notation, Definition

subspace, Theorem

linear combination, Definition Example Definition

linear transformation, Theorem

matrices, Example

system of equations, Example Example

linear combinations, Sage

solutions to linear systems, Theorem

linear dependence

more vectors than size, Theorem

linear independence, Definition Sage Definition

\(r\) and \(n\), Theorem

homogeneous systems, Theorem

injective linear transformation, Theorem

matrices, Example

orthogonal, Theorem

linear system

consistent, Theorem

matrix representation, Definition

notation, Definition

notation, Example

linear systems

notation, Example

linear transformation, Definition

addition, Definition Theorem Theorem

as matrix multiplication, Example

bases, Sage

basis of range, Example

checking, Example

composition, Definition Theorem Sage Sage

computing an inverse, Sage

defined by a matrix, Example

defined on a basis, Theorem Example Example Example

identity, Definition

injection, Definition

injective, Sage

inverse, Theorem Sage

inverse of inverse, Theorem

invertible, Definition Example

invertible, injective and surjective, Theorem

linear combination, Theorem

matrices, Sage

matrix of, Example Theorem Example

not, Example

not invertible, Example

notation, Definition

operations on, Sage

polynomials to matrices, Example

polynomials to polynomials, Example

rank plus nullity, Theorem

rank, nullity, Sage

restrictions, Sage

scalar multiple, Example

scalar multiplication, Definition

spanning range, Theorem

sum, Example

surjection, Definition

surjective, Sage

symbolic, Sage

vector space of, Theorem

zero vector, Theorem

linear transformation inverse

via matrix representation, Example

linear transformations

compositions, Example

from matrices, Theorem

linearly dependent

\(r\) and \(n\), Example

via homogeneous system, Example

linearly dependent columns

Archetype A, Example

linearly dependent set, Example

linear combinations within, Theorem

polynomials, Example

linearly independent

crazy vector space, Example

extending sets, Theorem

polynomials, Example

via homogeneous system, Example

linearly independent columns

Archetype B, Example

linearly independent set, Example Example

linearly independent spanning sets, Sage

lower triangular matrix, Definition

matrix, Definition Example

addition, Definition

notation, Definition

augmented, Definition

notation, Definition

column space, Definition

complex conjugate, Example

equality, Definition

notation, Definition

identity, Definition

inverse, Definition

nonsingular, Definition

notation, Definition

of a linear transformation, Theorem

product, Example Example

product with vector, Definition

row space, Definition

scalar multiplication, Definition

notation, Definition

square, Definition

submatrices, Example

submatrix, Definition

symmetric, Definition

transpose, Definition

unitary, Definition

unitary is invertible, Theorem

zero, Definition

matrix addition, Example

matrix creation, Sage

matrix entries

notation, Definition

matrix inverse, Sage

Archetype B, Example

computation, Theorem

nonsingular matrix, Theorem

of a matrix inverse, Theorem

one-sided, Theorem

product, Theorem

scalar multiple, Theorem

size 2 matrices, Theorem

transpose, Theorem

uniqueness, Theorem

matrix inverse, system of equations, Sage

matrix multiplication, Definition Sage

adjoints, Theorem

associativity, Theorem

complex conjugation, Theorem

distributivity, Theorem

entry-by-entry, Theorem

identity matrix, Theorem

inner product, Theorem

noncommutative, Example

notation, Definition

scalar matrix multiplication, Theorem

systems of linear equations, Theorem

transposes, Theorem

zero matrix, Theorem

matrix multiplication, properties, Sage

matrix operations, Sage

matrix product

as composition of linear transformations, Example

matrix representation, Definition Theorem

basis of eigenvectors, Example

change-of-basis, Sage

composition of linear transformations, Theorem

designing, Sage

invertible, Theorem

multiple of a linear transformation, Theorem

notation, Definition

sum of linear transformations, Theorem

upper triangular, Theorem

matrix representations, Example Sage

converting with change-of-basis, Example

matrix scalar multiplication, Example

matrix spaces, Sage

matrix vector space

dimension, Theorem

matrix-vector product, Example Sage

notation, Definition

more variables than equations, Theorem Example

multiplicative associativity

complex numbers, Property

multiplicative closure

complex numbers, Property

multiplicative commutativity

complex numbers, Property

multiplicative inverse

complex numbers, Property

nonsingular

columns as basis, Theorem

nonsingular matrices

linearly independent columns, Theorem

round 4, Sage

round 7, Sage

round 8, Sage

nonsingular matrices, round 2, Sage

nonsingular matrix, Sage

Archetype B, Example

column space, Theorem

elementary matrices, Theorem

equivalences, Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem

matrix inverse, Theorem

null space, Example

nullity, Theorem

product of nonsingular matrices, Theorem

rank, Theorem

row-reduced, Theorem

trivial null space, Theorem

unique solutions, Theorem

nonsingular matrix equivalences, Sage

round 5, Sage

round 6, Sage

nonsingular matrix equivalences, round 3, Sage

nonsingular matrix equivalences, round 9, Sage

nonsingular matrix, row-reduced, Example

norm, Example

inner product, Theorem

notation, Definition

normal matrix, Definition Example

orthonormal basis, Theorem

notation for a linear system, Example

Null space

as a span, Example

null space, Sage

Archetype I, Example

basis, Theorem

computation, Example Example

isomorphic to kernel, Theorem

linearly independent basis, Example

matrix, Definition

nonsingular matrix, Example

notation, Definition

singular matrix, Example

spanning set, Theorem Example Sage

subspace, Theorem

null space span, linearly independent

Archetype L, Example

nullity

computing, Theorem

injective linear transformation, Theorem

linear transformation, Definition

matrix, Definition Example

notation, Definition Definition

square matrix, Example

one

column vectors, Property

complex numbers, Property

matrices, Property

vectors, Property

orthogonal

linear independence, Theorem

set, Example

set of vectors, Definition

vector pairs, Definition

orthogonal vectors, Example

orthogonality and Gram-Schmidt, Sage

orthonormal, Definition

matrix columns, Example

orthonormal basis

normal matrix, Theorem

orthonormal diagonalization, Theorem

orthonormal set

four vectors, Example

three vectors, Example

particular solutions, Example

particular solutions, homogeneous solutions, Sage

polynomial

of a matrix, Example

polynomial vector space

dimension, Theorem

preimage, Definition

kernel, Theorem

preimages, Example Sage

product of triangular matrices, Theorem

properties of determinants, Sage

range

full, Example

isomorphic to column space, Theorem

linear transformation, Example

notation, Definition

of a linear transformation, Definition

preimage, Theorem

subspace, Theorem

surjective linear transformation, Theorem

via matrix representation, Example

rank

computing, Theorem

linear transformation, Definition

matrix, Definition Example

notation, Definition Definition

of transpose, Example

square matrix, Example

surjective linear transformation, Theorem

transpose, Theorem

rank+nullity, Theorem

rank, nullity of a matrix, Sage

reduced row-echelon form, Definition Example Example Sage

analysis

notation, Definition

extended, Definition

notation, Example

unique, Theorem

reducing a span, Example

relation of linear dependence, Definition Definition

relations of linear dependence, Sage

row operation

add

notation, Definition

multiply

notation, Definition

swap

notation, Definition

row operations, Definition Sage

elementary matrices, Example Theorem

row space, Sage

Archetype I, Example

as column space, Theorem

basis, Theorem Example

notation, Definition

row-equivalent matrices, Theorem

subspace, Theorem

row-equivalent matrices, Definition Example Theorem

row space, Theorem

row spaces, Example

row-reduce

symbolic matrix, Sage

row-reduced matrices, Theorem

Sage

getting started, Sage

sage under the hood

round 0, Sage

round 1, Sage

round 2, Sage

round 3, Sage

round 4, Sage

scalar closure

column vectors, Property

matrices, Property

vectors, Property

scalar multiple

matrix inverse, Theorem

scalar multiplication

zero scalar, Theorem

zero vector, Theorem

zero vector result, Theorem

scalar multiplication associativity

column vectors, Property

matrices, Property

vectors, Property

set, Definition

cardinality, Definition Example

notation, Definition

complement, Definition Example

notation, Definition

empty, Definition

equality, Definition

notation, Definition

intersection, Definition Example

notation, Definition

membership, Example

notation, Definition

union, Definition Example

notation, Definition

shoes, Theorem

similar matrices, Example Example Sage

equal eigenvalues, Theorem Example

similarity, Definition

equivalence relation, Theorem

singular matrix

Archetype A, Example

null space, Example

singular matrix, row-reduced, Example

socks, Theorem

solution set, Theorem

Archetype A, Example

archetype E, Example

solution set of a linear system, Definition

solution sets

possibilities, Theorem

solution to a linear system, Definition

solution vector, Definition

solutions, linear combinations, Sage

solving homogeneous system

Archetype A, Example

Archetype B, Example

Archetype D, Example

solving linear systems, Sage Sage

solving nonlinear equations, Example

solving systems, Sage

span, Definition Definition

basic, Example

basis, Theorem

casting out vectors, Sage

improved, Example

notation, Definition

reduced, Sage

reducing, Example

reduction, Example

removing vectors, Example

reworked, Sage

reworking elements, Example

set of polynomials, Example

subspace, Theorem

span of columns

Archetype A, Example

Archetype B, Example

Archetype D, Example

spanning set, Definition

crazy vector space, Example

matrices, Example

more vectors, Theorem

polynomials, Example

spanning sets, Sage

standard unit vector

notation, Definition

submatrix

notation, Definition

subset, Definition

notation, Definition

subspace, Definition

as null space, Example

characterized, Example

in \(P_4\), Example

intersection, Theorem

not, additive closure, Example

not, scalar closure, Example

not, zero vector, Example

sum, Definition Definition Theorem

sum, intersection dimensions, Theorem

testing, Theorem

trivial, Definition

verification, Example Example

subspaces

equal dimension, Theorem

surjective, Example

Archetype N, Example

not, Example Example

not, Archetype O, Example

not, by dimension, Example

polynomials to matrices, Example

surjective linear transformation

bases, Theorem

surjective linear transformations

dimension, Theorem

symmetric matrices, Theorem

symmetric matrix, Example

system of equations

vector equality, Example

system of linear equations, Definition

trail mix, Example

transpose, Example

matrix addition, Theorem

matrix inverse, Theorem Theorem

matrix scalar multiplication, Theorem

notation, Definition

scalar multiplication, Theorem

transpose of a transpose, Theorem

triangular matrix

inverse, Theorem

trivial solution

system of equations, Definition

typical systems, \(2\times 2\), Example

unique solution, \(3\times 3\), Example Example

unit vectors, Definition

basis, Theorem

orthogonal, Example

unitary

permutation matrix, Example

size 3, Example

unitary matrices, Sage

columns, Theorem

unitary matrix

inner product, Theorem

upper triangular matrix, Definition

vector

addition, Definition

column, Definition

equality, Definition

notation, Definition

inner product, Definition

norm, Definition

notation, Definition

of constants, Definition

scalar multiplication, Definition

vector addition, Example

vector creation, Sage

vector entries

notation, Definition

vector form of solutions, Example Theorem

Archetype D, Example

Archetype I, Example

Archetype L, Example

vector operations, Sage

vector representation, Example Theorem Example

injective, Theorem

invertible, Theorem

linear transformation, Definition Theorem

notation, Definition

surjective, Theorem

vector representations, Sage

polynomials, Example

vector scalar multiplication, Example

vector space, Definition

characterization, Theorem

column vectors, Definition

infinite dimension, Example

linear transformations, Theorem

vector space of column vectors

notation, Definition

vector space of functions, Example

vector space of infinite sequences, Example

vector space of matrices, Definition Example

notation, Definition

vector space of polynomials, Example

vector space properties

column vectors, Theorem

matrices, Theorem

vector space, crazy, Example

vector space, singleton, Example

vector spaces, Sage

column vectors, Sage

isomorphic, Definition Theorem

zero

complex numbers, Property

zero column vector, Definition

notation, Definition

zero matrix

notation, Definition

zero vector

column vectors, Property

matrices, Property

unique, Theorem

vectors, Property