Skip to main content
\(\newcommand{\orderof}[1]{\sim #1} \newcommand{\Z}{\mathbb{Z}} \newcommand{\reals}{\mathbb{R}} \newcommand{\real}[1]{\mathbb{R}^{#1}} \newcommand{\complexes}{\mathbb{C}} \newcommand{\complex}[1]{\mathbb{C}^{#1}} \newcommand{\conjugate}[1]{\overline{#1}} \newcommand{\modulus}[1]{\left\lvert#1\right\rvert} \newcommand{\zerovector}{\vect{0}} \newcommand{\zeromatrix}{\mathcal{O}} \newcommand{\innerproduct}[2]{\left\langle#1,\,#2\right\rangle} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\dimension}[1]{\dim\left(#1\right)} \newcommand{\nullity}[1]{n\left(#1\right)} \newcommand{\rank}[1]{r\left(#1\right)} \newcommand{\ds}{\oplus} \newcommand{\detname}[1]{\det\left(#1\right)} \newcommand{\detbars}[1]{\left\lvert#1\right\rvert} \newcommand{\trace}[1]{t\left(#1\right)} \newcommand{\sr}[1]{#1^{1/2}} \newcommand{\spn}[1]{\left\langle#1\right\rangle} \newcommand{\nsp}[1]{\mathcal{N}\!\left(#1\right)} \newcommand{\csp}[1]{\mathcal{C}\!\left(#1\right)} \newcommand{\rsp}[1]{\mathcal{R}\!\left(#1\right)} \newcommand{\lns}[1]{\mathcal{L}\!\left(#1\right)} \newcommand{\per}[1]{#1^\perp} \newcommand{\augmented}[2]{\left\lbrack\left.#1\,\right\rvert\,#2\right\rbrack} \newcommand{\linearsystem}[2]{\mathcal{LS}\!\left(#1,\,#2\right)} \newcommand{\homosystem}[1]{\linearsystem{#1}{\zerovector}} \newcommand{\rowopswap}[2]{R_{#1}\leftrightarrow R_{#2}} \newcommand{\rowopmult}[2]{#1R_{#2}} \newcommand{\rowopadd}[3]{#1R_{#2}+R_{#3}} \newcommand{\leading}[1]{\boxed{#1}} \newcommand{\rref}{\xrightarrow{\text{RREF}}} \newcommand{\elemswap}[2]{E_{#1,#2}} \newcommand{\elemmult}[2]{E_{#2}\left(#1\right)} \newcommand{\elemadd}[3]{E_{#2,#3}\left(#1\right)} \newcommand{\scalarlist}[2]{{#1}_{1},\,{#1}_{2},\,{#1}_{3},\,\ldots,\,{#1}_{#2}} \newcommand{\vect}[1]{\mathbf{#1}} \newcommand{\colvector}[1]{\begin{bmatrix}#1\end{bmatrix}} \newcommand{\vectorcomponents}[2]{\colvector{#1_{1}\\#1_{2}\\#1_{3}\\\vdots\\#1_{#2}}} \newcommand{\vectorlist}[2]{\vect{#1}_{1},\,\vect{#1}_{2},\,\vect{#1}_{3},\,\ldots,\,\vect{#1}_{#2}} \newcommand{\vectorentry}[2]{\left\lbrack#1\right\rbrack_{#2}} \newcommand{\matrixentry}[2]{\left\lbrack#1\right\rbrack_{#2}} \newcommand{\lincombo}[3]{#1_{1}\vect{#2}_{1}+#1_{2}\vect{#2}_{2}+#1_{3}\vect{#2}_{3}+\cdots +#1_{#3}\vect{#2}_{#3}} \newcommand{\matrixcolumns}[2]{\left\lbrack\vect{#1}_{1}|\vect{#1}_{2}|\vect{#1}_{3}|\ldots|\vect{#1}_{#2}\right\rbrack} \newcommand{\transpose}[1]{#1^{t}} \newcommand{\inverse}[1]{#1^{-1}} \newcommand{\submatrix}[3]{#1\left(#2|#3\right)} \newcommand{\adj}[1]{\transpose{\left(\conjugate{#1}\right)}} \newcommand{\adjoint}[1]{#1^\ast} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\setparts}[2]{\left\lbrace#1\,\middle|\,#2\right\rbrace} \newcommand{\card}[1]{\left\lvert#1\right\rvert} \newcommand{\setcomplement}[1]{\overline{#1}} \newcommand{\charpoly}[2]{p_{#1}\left(#2\right)} \newcommand{\eigenspace}[2]{\mathcal{E}_{#1}\left(#2\right)} \newcommand{\eigensystem}[3]{\lambda&=#2&\eigenspace{#1}{#2}&=\spn{\set{#3}}} \newcommand{\geneigenspace}[2]{\mathcal{G}_{#1}\left(#2\right)} \newcommand{\algmult}[2]{\alpha_{#1}\left(#2\right)} \newcommand{\geomult}[2]{\gamma_{#1}\left(#2\right)} \newcommand{\indx}[2]{\iota_{#1}\left(#2\right)} \newcommand{\ltdefn}[3]{#1\colon #2\rightarrow#3} \newcommand{\lteval}[2]{#1\left(#2\right)} \newcommand{\ltinverse}[1]{#1^{-1}} \newcommand{\restrict}[2]{{#1}|_{#2}} \newcommand{\preimage}[2]{#1^{-1}\left(#2\right)} \newcommand{\rng}[1]{\mathcal{R}\!\left(#1\right)} \newcommand{\krn}[1]{\mathcal{K}\!\left(#1\right)} \newcommand{\compose}[2]{{#1}\circ{#2}} \newcommand{\vslt}[2]{\mathcal{LT}\left(#1,\,#2\right)} \newcommand{\isomorphic}{\cong} \newcommand{\similar}[2]{\inverse{#2}#1#2} \newcommand{\vectrepname}[1]{\rho_{#1}} \newcommand{\vectrep}[2]{\lteval{\vectrepname{#1}}{#2}} \newcommand{\vectrepinvname}[1]{\ltinverse{\vectrepname{#1}}} \newcommand{\vectrepinv}[2]{\lteval{\ltinverse{\vectrepname{#1}}}{#2}} \newcommand{\matrixrep}[3]{M^{#1}_{#2,#3}} \newcommand{\matrixrepcolumns}[4]{\left\lbrack \left.\vectrep{#2}{\lteval{#1}{\vect{#3}_{1}}}\right|\left.\vectrep{#2}{\lteval{#1}{\vect{#3}_{2}}}\right|\left.\vectrep{#2}{\lteval{#1}{\vect{#3}_{3}}}\right|\ldots\left|\vectrep{#2}{\lteval{#1}{\vect{#3}_{#4}}}\right.\right\rbrack} \newcommand{\cbm}[2]{C_{#1,#2}} \newcommand{\jordan}[2]{J_{#1}\left(#2\right)} \newcommand{\hadamard}[2]{#1\circ #2} \newcommand{\hadamardidentity}[1]{J_{#1}} \newcommand{\hadamardinverse}[1]{\widehat{#1}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

IndexIndex

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
augmented matrix, Example
column space, Example
linearly dependent columns, Example
singular matrix, Example
solving homogeneous system, Example
system as linear combination, Example
Archetype B
column space, Example
inverse, Example
linearly independent columns, Example
nonsingular matrix, Example
not invertible, Example
solutions, Example
solutions via inverse, Example
solving homogeneous system, Example
system as linear combination, Example
vector equality, 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
as null space, Theorem
Archetype A, Example
Archetype B, Example
as null space, Example
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, Example
modulus, Example
complex number
conjugate, Definition
modulus, Definition
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
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, 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
pre-image, 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
system of equations, Example
linear transformation, Theorem
matrices, Example
system of equations, 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
polynomials to polynomials, Example
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
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
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
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
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, Sage
Archetype I, Example
as a span, 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
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
pre-image, Definition
kernel, Theorem
pre-images, 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
pre-image, Theorem
subspace, Theorem
surjective linear transformation, Theorem
via matrix representation, Example
rank
computing, Theorem
linear transformation, 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
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
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
notation, Definition
shoes, Theorem
similar matrices, Example, Example, Sage
equal eigenvalues, Example
eual eigenvalues, Theorem
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
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
not, additive closure, Example
not, scalar closure, Example
not, zero vector, Example
testing, Theorem
trivial, Definition
verification, Example, Example
subspaces
equal dimension, Theorem
surjective, Example
Archetype N, 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 scalar multiplication, Theorem
matrix addition, Theorem
matrix inverse, Theorem, 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
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