## IndexIndex

column vectors
complex numbers
matrices
vectors
column vectors
complex numbers
matrices
vectors
complex numbers
complex numbers
from scalar multiplication
column vectors
matrices
unique
vectors
inner product
notation
of a matrix sum
of matrix scalar multiplication
Archetype A
column space
linearly dependent columns
singular matrix
solving homogeneous system
system as linear combination
archetype A
augmented matrix
Archetype B
column space
inverse
linearly independent columns
nonsingular matrix
not invertible
solutions via inverse
solving homogeneous system
system as linear combination
vector equality
archetype B
solutions
Archetype C
homogeneous system
Archetype D
column space, original columns
solving homogeneous system
vector form of solutions
Archetype I
casting out vectors
column space from row operations
null space
row space
vector form of solutions
Archetype L
null space span, linearly independent
vector form of solutions
augmented matrix, Sage
bases, Sage
basis
columns nonsingular matrix
common size
crazy vector space
matrices
polynomials
subspace of matrices
best cities
money magazine
change of basis
between polynomials
change-of-basis
between column vectors
matrix representation
similarity
change-of-basis matrix
inverse
characteristic polynomial
degree
size 3 matrix
coefficient matrix
nonsingular
coercion, Sage
column space
Archetype A
Archetype B
as null space
as null space, Archetype G
as row space
basis
consistent system
consistent systems
isomorphic to range
nonsingular matrix
notation
original columns, Sage
original columns, Archetype D
row operations, Archetype I
subspace
testing membership
two computations
notation
column vector scalar multiplication
notation
commutativity
column vectors
matrices
vectors
complex $m$-space
complex arithmetic
complex number
conjugate
modulus
complex numbers
notation
arithmetic properties
equality
notation
multiplication
notation
zero product
complex vector space
dimension
composition
injective linear transformations
surjective linear transformations
conjugate
column vector
matrix
notation
multiplication
notation
of conjugate of a matrix
scalar multiplication
twice
conjugate of a vector
notation
conjugates, norms, inner products, Sage
conjugation
matrix scalar multiplication
matrix transpose
consistent linear system
consistent linear systems
consistent system
consistent systems, spans, Sage
coordinates, Sage
orthonormal basis
coordinatization
linear combination of matrices
linear independence
orthonormal basis
spanning sets
coordinatizing
polynomials
crazy vector space
properties
determinant
bars
notation
computed two ways
equal rows or columns
expansion, columns
expansion, rows
functional
notation
identity matrix
matrix multiplication
nonsingular matrix
row or column multiple
row or column swap
size 2 matrix
size 3 matrix
transpose
via row operations
zero
zero row or column
zero versus nonzero
determinant, upper triangular matrix
determinants
elementary matrices
diagonal matrix
diagonalizable
distinct eigenvalues
full eigenspaces
not
diagonalizable matrix
high power
diagonalization
Archetype B
criteria
diagonalization of a matrix, Sage
dimension
crazy vector space
notation
polynomial subspace
proper subspaces
subspace
dimensions
matrix subspaces, Sage
distributivity
complex numbers
matrices
column vectors
matrices
vectors
column vectors
vectors
eigenspace
as null space
subspace
eigenvalue
algebraic multiplicity
notation
complex
existence
geometric multiplicity
notation
linear transformation
multiplicities
power
root of characteristic polynomial
scalar multiple
singular matrix
symmetric matrix
zero
eigenvalues
building desired
complex, of a linear transformation
computing, Sage
conjugate pairs
distinct
Hermitian matrices
inverse
maximum number
multiplicities
number
of a polynomial
size 3 matrix
transpose
eigenvalues and eigenvectors, Sage
eigenvalues, eigenvectors
vector, matrix representations
eigenvalues}\index{eigenvectors
eigenvectors
computing, Sage
conjugate pairs
Hermitian matrices
linear transformation
linearly independent
of a linear transformation
elementary matrices
determinants
nonsingular
row operations
elementary matrix
notation
multiply
notation
swap
notation
empty set
notation
endomorphisms, Sage
equal matrices
via equal matrix-vector products
equation operations
equivalent systems
exact vs.\ inexact computations, Sage
extended echelon form, Sage
submatrices
extended reduced row-echelon form
properties
Fibonacci sequence
four subsets
four subspaces
dimension
free variables
free variables, number
free, dependent variables, Sage
free, independent variables
goldilocks
Gram-Schmidt
column vectors
three vectors
help, Sage
hermitian
Hermitian matrix
inner product
homogeneous system
Archetype C
consistent
infinitely many solutions
homogeneous systems
linear independence
solving, Sage
identity matrix
determinant
notation
independent, dependent variables
indesxstring
indexstring
infinite solution set
infinite solutions, $3\times 4$
injective
not
not, by dimension
polynomials to matrices
injective linear transformation
bases
injective linear transformations
dimension
inner product
anti-commutative
norm
notation
positive
scalar multiplication
inverse
composition of linear transformations
notation
invertible linear transformation
defined by invertible matrix
invertible linear transformations
composition
computing
isomorphic
multiple vector spaces
vector spaces
isomorphic vector spaces
dimension
kernel
injective linear transformation
isomorphic to null space
linear transformation
notation
of a linear transformation
preimage
subspace
trivial
via matrix representation
left null space
as row space
notation
subspace
linear combination
linear transformation
matrices
system of equations
linear combinations, Sage
solutions to linear systems
linear dependence
more vectors than size
linear independence
$r$ and $n$
homogeneous systems
injective linear transformation
matrices
orthogonal
linear system
consistent
matrix representation
notation
notation
linear systems
notation
linear transformation
as matrix multiplication
bases, Sage
basis of range
checking
composition
computing an inverse, Sage
defined by a matrix
defined on a basis
identity
injection
injective, Sage
inverse
inverse of inverse
invertible
invertible, injective and surjective
linear combination
matrices, Sage
matrix of
not
not invertible
notation
operations on, Sage
polynomials to matrices
polynomials to polynomials
rank plus nullity
rank, nullity, Sage
restrictions, Sage
scalar multiple
scalar multiplication
spanning range
sum
surjection
surjective, Sage
symbolic, Sage
vector space of
zero vector
linear transformation inverse
via matrix representation
linear transformations
compositions
from matrices
linearly dependent
$r$ and $n$
via homogeneous system
linearly dependent columns
Archetype A
linearly dependent set
linear combinations within
polynomials
linearly independent
crazy vector space
extending sets
polynomials
via homogeneous system
linearly independent columns
Archetype B
linearly independent set
linearly independent spanning sets, Sage
lower triangular matrix
matrix
notation
augmented
notation
column space
complex conjugate
equality
notation
identity
inverse
nonsingular
notation
of a linear transformation
product
product with vector
row space
scalar multiplication
notation
square
submatrices
submatrix
symmetric
transpose
unitary
unitary is invertible
zero
matrix creation, Sage
matrix entries
notation
matrix inverse, Sage
Archetype B
computation
nonsingular matrix
of a matrix inverse
one-sided
product
scalar multiple
size 2 matrices
transpose
uniqueness
matrix inverse, system of equations, Sage
matrix multiplication
associativity
complex conjugation
distributivity
entry-by-entry
identity matrix
inner product
noncommutative
notation
scalar matrix multiplication
systems of linear equations
transposes
zero matrix
matrix multiplication, properties, Sage
matrix operations, Sage
matrix product
as composition of linear transformations
matrix representation
basis of eigenvectors
change-of-basis, Sage
composition of linear transformations
designing, Sage
invertible
multiple of a linear transformation
notation
sum of linear transformations
upper triangular
matrix representations
converting with change-of-basis
matrix scalar multiplication
matrix spaces, Sage
matrix vector space
dimension
matrix-vector product
notation
more variables than equations
multiplicative associativity
complex numbers
multiplicative closure
complex numbers
multiplicative commutativity
complex numbers
multiplicative inverse
complex numbers
nonsingular
columns as basis
nonsingular matrices
linearly independent columns
round 4, Sage
round 7, Sage
round 8, Sage
nonsingular matrices, round 2, Sage
nonsingular matrix, Sage
Archetype B
column space
elementary matrices
equivalences
matrix inverse
null space
nullity
product of nonsingular matrices
rank
row-reduced
trivial null space
unique solutions
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
norm
inner product
notation
normal matrix
orthonormal basis
notation for a linear system
Null space
as a span
null space, Sage
Archetype I
basis
computation
isomorphic to kernel
linearly independent basis
matrix
nonsingular matrix
notation
singular matrix
spanning set
subspace
null space span, linearly independent
Archetype L
nullity
computing
injective linear transformation
linear transformation
matrix
notation
square matrix
one
column vectors
complex numbers
matrices
vectors
orthogonal
linear independence
set
set of vectors
vector pairs
orthogonal vectors
orthogonality and Gram-Schmidt, Sage
orthonormal
matrix columns
orthonormal basis
normal matrix
orthonormal diagonalization
orthonormal set
four vectors
three vectors
particular solutions
particular solutions, homogeneous solutions, Sage
polynomial
of a matrix
polynomial vector space
dimension
preimage
kernel
preimages
product of triangular matrices
properties of determinants, Sage
range
full
isomorphic to column space
linear transformation
notation
of a linear transformation
preimage
subspace
surjective linear transformation
via matrix representation
rank
computing
linear transformation
matrix
notation
of transpose
square matrix
surjective linear transformation
transpose
rank+nullity
rank, nullity of a matrix, Sage
reduced row-echelon form
analysis
notation
extended
notation
unique
reducing a span
relation of linear dependence
relations of linear dependence, Sage
row operation
notation
multiply
notation
swap
notation
row operations
elementary matrices
row space, Sage
Archetype I
as column space
basis
notation
row-equivalent matrices
subspace
row-equivalent matrices
row space
row spaces
row-reduce
symbolic matrix, Sage
row-reduced matrices
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
matrices
vectors
scalar multiple
matrix inverse
scalar multiplication
zero scalar
zero vector
zero vector result
scalar multiplication associativity
column vectors
matrices
vectors
set
cardinality
notation
complement
notation
empty
equality
notation
intersection
notation
membership
notation
union
notation
shoes
similar matrices
equal eigenvalues
similarity
equivalence relation
singular matrix
Archetype A
null space
singular matrix, row-reduced
socks
solution set
Archetype A
archetype E
solution set of a linear system
solution sets
possibilities
solution to a linear system
solution vector
solutions, linear combinations, Sage
solving homogeneous system
Archetype A
Archetype B
Archetype D
solving linear systems
solving nonlinear equations
solving systems, Sage
span
basic
basis
casting out vectors, Sage
improved
notation
reduced, Sage
reducing
reduction
removing vectors
reworked, Sage
reworking elements
set of polynomials
subspace
span of columns
Archetype A
Archetype B
Archetype D
spanning set
crazy vector space
matrices
more vectors
polynomials
spanning sets, Sage
standard unit vector
notation
submatrix
notation
subset
notation
subspace
as null space
characterized
in $P_4$
intersection
not, scalar closure
not, zero vector
sum
sum, intersection dimensions
testing
trivial
verification
subspaces
equal dimension
surjective
Archetype N
not
not, Archetype O
not, by dimension
polynomials to matrices
surjective linear transformation
bases
surjective linear transformations
dimension
symmetric matrices
symmetric matrix
system of equations
vector equality
system of linear equations
trail mix
transpose
matrix inverse
matrix scalar multiplication
notation
scalar multiplication
transpose of a transpose
triangular matrix
inverse
trivial solution
system of equations
typical systems, $2\times 2$
unique solution, $3\times 3$
unit vectors
basis
orthogonal
unitary
permutation matrix
size 3
unitary matrices, Sage
columns
unitary matrix
inner product
upper triangular matrix
vector
column
equality
notation
inner product
norm
notation
of constants
scalar multiplication
vector creation, Sage
vector entries
notation
vector form of solutions
Archetype D
Archetype I
Archetype L
vector operations, Sage
vector representation
injective
invertible
linear transformation
notation
surjective
vector representations, Sage
polynomials
vector scalar multiplication
vector space
characterization
column vectors
infinite dimension
linear transformations
vector space of column vectors
notation
vector space of functions
vector space of infinite sequences
vector space of matrices
notation
vector space of polynomials
vector space properties
column vectors
matrices
vector space, crazy
vector space, singleton
vector spaces, Sage
column vectors, Sage
isomorphic
zero
complex numbers
zero column vector
notation
zero matrix
notation
zero vector
column vectors
matrices
unique
vectors