IndexIndex

-coordinates

change of basis matrixImportant Note

computing

row reductionImportant Note

with respect to an orthogonal basisImportant Note

definition ofDefinition

informallyParagraph

labeling pointsImportant Note

nonstandard gridExample

Algebraic multiplicity

and diagonalizabilityTheorem

and geometric multiplicityTheorem

definition ofDefinition

equals oneImportant Note

of similar matricesTheorem

Approximate solutionsee Least-squares

Augmented matrixsee Matrix

Basis

and orthogonal projectionCorollary

basis theoremBasis Theorem

coordinates with respect tosee -coordinates

definition ofDefinition

infinitely manyImportant Note

making orthogonalTheorem

of Example Example

of a column spaceTheorem

of a null spaceTheorem

of a spanParagraphs

span of an orthogonal setFact

uniqueness with respect toFact

Best-fit problemSubsection

best-fit ellipseExample

best-fit lineExample

best-fit linear equationExample

best-fit parabolaExample

best-fit trigonometric functionExample

general setupParagraph

Block Diagonalization TheoremBlock Diagonalization Theorem

Characteristic polynomial

and eigenvaluesTheorem

definition ofDefinition

factoring by handParagraphs

form ofTheorem

of a matrixImportant Note

of similar matricesFact

Codomainsee Transformation

Cofactorsee Matrix

Color spaceExample

Column ranksee Rank

Column space

and invertibilityInvertible Matrix Theorem

and ranksee Rank

basis ofsee Basis

definition ofDefinition

is a subspaceParagraph

is row space of transposeParagraph

of an orthogonal projectionProposition

orthogonal complement ofProposition Important Note

range of a transformationImportant Note

versus the solution setSubsection

Column spansee Column space

Complex conjugationsee Complex numbers

Complex eigenvalue

matrices

and rotation-scaling matricesRotation-Scaling Theorem

and rotation-scaling matrices, computingImportant Note

different rotation-scaling matricesParagraph

dynamics ofNote Example Example Example

geometry ofSubsection

matrices

dynamics ofExample

conjugate pairsImportant Note

existence ofImportant Note

Complex numbers

absolute valueItem

arithmetic ofNote

conjugationItem

definition ofDefinition

real and imaginary parts ofItem

vectorsParagraph

Consistentsee System of linear equations

Cramer’s ruleTheorem

and computing inversesTheorem

Determinant

alternative defining properties ofRemark

and column operationsCorollary

and computing inversesTheorem

and powers of matricesCorollary

and row operationsDefinition

and volumesTheorem

computation of

cofactor expansionTheorem

row reductionRecipe

defining properties ofDefinition

existence and uniqueness ofTheorem

identity matrixDefinition

invertibility propertyProposition Invertible Matrix Theorem

methods of computationImportant Note

multilinearity propertyProposition

multiplicativity propertyProposition

and volumesRemark

of a matrixDefinition Example Example

of a matrixImportant Note

of similar matricesParagraph

properties ofImportant Note

transpose propertyProposition

Diagonal

see MatrixDefinition

Diagonalizability

algebraic-geometric multiplicity criterionTheorem

criterionDiagonalization Theorem

criterion for similarityImportant Note

definition ofDefinition

diagonal matricesExample Example

distinct eigenvaluesImportant Note

dynamics ofExample Example Example Example

geometry ofSubsection

is unrelated to invertibilityImportant Note

of matricesExample

of a projection matrixProposition

order of eigenvaluesNote

powers ofImportant Note

projectionExample

recipeImportant Note

shearExample

similar matricesExample

Difference equationDefinition

DilationExample Example Example

eigenvectors ofExample

inverse ofExample Example

Dimension

definition ofDefinition

of a column spaceDefinition

of a null spaceDefinition

of a solution setImportant Note Important Note

of an orthogonal complementFact

Domainsee Transformation

Dot product

and anglesRemark

and distanceDefinition

and lengthFact

definition ofDefinition

properties ofNote

Eigenspace

and the null spaceFact

computationImportant Note

definition ofDefinition

is a solution setParagraph

is a subspaceParagraph

of a projection matrixProposition

of similar matricesImportant Note

orthogonal complement ofExample

Eigenvalue

algebraic multiplicity ofsee Algebraic multiplicity

and diagonalizabilityImportant Note

and invertibilityInvertible Matrix Theorem

and stochastic matricesFact

and the characteristic polynomialsee Characteristic polynomial

complexsee Complex eigenvalue

definition ofDefinition

eigenvector forDefinition

geometric multiplicity ofsee Geometric multiplicity

identity matrixExample

maximum number ofImportant Note

of a projection matrixProposition

of a triangular matrixCorollary

of similar matricesImportant Note

zeroFact

Eigenvector

and collinearityParagraph

and diagonalizabilityDiagonalization Theorem

and stochastic matricesDefinition

computationImportant Note

trick for matricesNote

definition ofDefinition

eigenvalue forDefinition

identity matrixExample

linear independence ofFact

of a projection matrixProposition

of similar matricesFact

Elimination methodSubsection

Ellipse

area ofsee Volume

best-fit problemExample

Equation of linear dependencesee Linear independence

Free variableDefinition

Functionsee Transformation

Fundamental theorem of algebraFundamental Theorem of Algebra

Gaussian eliminationsee Row reduction

Geometric multiplicity

and algebraic multiplicityTheorem

and diagonalizabilityTheorem

definition ofDefinition

of similar matricesTheorem

Google PageRank

Google MatrixDefinition

eigenvector ofImportant Note

importance matrixDefinition

eigenvector ofImportant Note

importance ruleDefinition

Gram–Schmidt ProcessTheorem

detecting linear dependenceImportant Note

Homogeneoussee System of linear equations

Identity matrix

and identity transformationExample

and matrix multiplicationNote

and standard coordinate vectorsDefinition

as a functionExample

as a matrix transformationExample

definition ofDefinition

determinant ofDefinition

eigenvalues ofExample

eigenvectors ofExample

similarityParagraph

Identity transformation

and compositionNote

and identity matrixExample

definition ofDefinition

Imaginary numbersee Complex numbers

Imaginary partsee Complex numbers

Implicit equationSubsection Note

Inconsistentsee System of linear equations

Increasing span criterionsee Linear independence

Inhomogeneoussee System of linear equations

Invertible matrix

and invertible transformationTheorem

basic factsFact

computation

caseProposition

in generalTheorem

using Cramer’ ruleTheorem

definition ofDefinition

determinant ofProposition Invertible Matrix Theorem

inverse ofItem

invertible matrix theoremInvertible Matrix Theorem

solving linear systems withTheorem

Invertible matrix theoremInvertible Matrix Theorem Invertible Matrix Theorem

Invertible transformation

and invertible matricesTheorem Invertible Matrix Theorem

definition ofDefinition

one-to-one and ontoProposition

Least-squares

and Important Note

computation of

complicated matrix formulaTheorem

Projection FormulaImportant Note

row reductionTheorem Important Note

definition ofDefinition

picture ofParagraphs

uniqueness ofTheorem

Line

dimension-1 solution setImportant Note Important Note

geometric definition ofDefinition

number lineExample

orthogonal projection ontoImportant Note Example

parametric form ofExample

Linear combination

collinear vectors, picture ofExample

definition ofDefinition

single vector, picture ofExample

two vectors, picture ofExample

Linear dependencesee Linear independence

Linear dependence relationsee Linear independence

Linear equation

definition ofDefinition

system ofsee System of linear equations

Linear independence

and determinantsCorollary

and invertibilityInvertible Matrix Theorem

basic factsFact

definition ofDefinition

equation of linear dependenceParagraph

increasing span criterionTheorem

linear dependence relationParagraph

of an orthogonal setFact

pictures ofSubsection

verifyingImportant Note

with Gram–SchmidtImportant Note

wide matricesImportant Note

Linear transformation

addition ofsee Transformation

and volumessee Matrix transformation

are matrix transformationsImportant Note

basic factsFact

composition ofsee Transformation

and matrix multiplicationTheorem

linearity ofTheorem

definition ofDefinition

dictionaryNote

invertiblesee Invertible transformation

scalar multiplication ofsee Transformation

standard matrix ofTheorem

orthogonal projectionParagraph

verifying nonlinearityExample

when defined by a formulaParagraph

Lower-triangular

see MatrixDefinition

Matrix

addition ofDefinition

as a functionsee Matrix transformationParagraph

augmentedParagraphs

cofactor ofDefinition

and determinantsTheorem

sign ofParagraph

definition ofParagraphs

determinant ofsee Determinant

diagonal entries ofDefinition

inverse ofsee Invertible matrix

invertiblesee Invertible matrix

lower-triangularDefinition

determinant ofProposition

eigenvalues ofCorollary

minor ofDefinition

multiplicationsee Matrix multiplication

nullity ofsee Nullity

parallelepiped determined byTheorem

product with vectorsee Matrix-vector product

projectionsee Orthogonal projection, standard matrix of

rank ofsee Rank

rotation-scalingsee Rotation-scaling matrix

scalar multiplication ofDefinition

similarsee Similarity

size ofImportant Note

stochasticsee Stochastic matrix

trace ofDefinition

similar matricesParagraph

transpose ofDefinition

and productsFact

determinant ofProposition

upper-triangularDefinition

determinant ofProposition

eigenvalues ofCorollary

Matrix equation

always consistentTheorem

and invertibilityInvertible Matrix Theorem

definition ofDefinition

equivalence with vector equationNote

solving with the inverse matrixTheorem

spans and consistencyNote

Matrix multiplication

and composition of transformationsTheorem

and the matrix-vector productParagraph

associativity ofNote

caveatsImportant Note

definition ofDefinition

determinant ofProposition

inverse ofItem

noncommutativity ofExample

order of operationsExample

powersParagraph

and diagonalizabilityImportant Note

and similarityFact

properties ofNote

row-column ruleImportant Note

size of matricesImportant Note

Matrix transformation

addition ofsee Transformation

and volumesTheorem

codomain ofImportant Note

composition ofsee Transformationsee Linear transformation

definition ofDefinition

dictionaryNote

domain ofImportant Note

invertiblesee Invertible transformation

linearity ofParagraph

of Example

one-to-one criteriaTheorem

onto criteriaTheorem

range ofImportant Note

scalar multiplication ofsee Transformation

tall matricesNote

wide matricesNote

Matrix-vector product

and matrix multiplicationParagraph

definition ofDefinition

row-column ruleImportant Note

with standard coordinate vectorsFact

Minorsee Matrix

Multiplicity

algebraicsee Algebraic multiplicity

geometricsee Geometric multiplicity

Nontrivial solutionsee System of linear equations

Null space

and invertibilityInvertible Matrix Theorem

basis ofsee Basis

computingImportant Note

definition ofDefinition

is a solution setExample

is a subspaceSubsection

is the -eigenspaceFact

of an orthogonal projectionProposition

orthogonal complement ofImportant Note

NullityDefinition

rank theoremRank Theorem

One-to-one

and invertibilityInvertible Matrix Theorem

criteria for matrix transformationsTheorem

definition ofDefinition

equivalent formulationsParagraph

finding two vectors with the same imageParagraph

functions of one variableExample

negation ofParagraph

square matricesNote

versus ontoParagraph

wide matricesNote

Onto

and invertibilityInvertible Matrix Theorem

criteria for matrix transformationsTheorem

definition ofDefinition

equivalent formulationsParagraph

finding a vector not in the rangeParagraph

functions of one variableExample

negation ofParagraph

square matricesNote

tall matricesNote

versus one-to-oneParagraph

Orthogonal complement

basic factsFact

computation ofSubsection

definition ofDefinition

dimension ofFact

of a column spaceProposition Important Note

of a null spaceImportant Note

of a row spaceImportant Note

of a spanParagraph Important Note

of an eigenspaceExample

orthogonal complement ofFact

pictures ofParagraphs

system of linear equationsParagraph

Orthogonal decompositionsee Orthogonal projection

Orthogonal projection

and -coordinatesImportant Note

as a transformationSubsection

composed with itselfProposition

computation of

complicated matrix formulaCorollary

Projection FormulaProjection Formula

row reductionTheorem Important Note

definition ofDefinition

distance fromImportant Note

eigenvectors ofExample

existence ofTheorem

is the closest vectorImportant Note

linearity ofProposition

of a vector in Example

onto a column spaceImportant Note

onto a lineImportant Note Example

onto the -planeExample Example Example

noninvertibility ofNon-Example

properties ofProposition

range ofProposition

standard matrix ofParagraph

column space ofProposition

complicated matrix formulaImportant Note

diagonalizability ofProposition

eigenvalues ofProposition

eigenvectors ofProposition

noninvertibility ofProposition

null space ofProposition

properties ofProposition

square ofProposition

Orthogonal set

and -coordinatesImportant Note

and least squaresImportant Note

definition ofDefinition

linear independence ofFact

making orthonormalImportant Note

necessity ofImportant Note

orthonormality ofDefinition

standard coordinate vectorsExample

producing from a basisTheorem

Orthogonality

and the Pythagorean theoremImportant Note

definition ofDefinition

zero vectorImportant Note

Orthonormal setsee Orthogonal set

Parallelepiped

definition ofDefinition

flatKey Observation

parallelogramExample

area ofExample

volume ofTheorem

Parallelogramsee Parallelepiped

Parameterized equationNote

Parametric formSubsection Important Note

Parametric vector form

of a homogeneous equationExample Important Note

of an inhomogeneous equationParagraph

particular solutionKey Observation

Particular solutionsee Parametric vector form

Perron–Frobenius theoremTheorem

PivotDefinition

Pivot positionDefinition

Plane

dimension-2 solution setImportant Note Important Note

geometric definition ofDefinition

parametric form ofExample

-planeExample

PointDefinition

distance betweenDefinition

Polynomial

characteristicsee Characteristic polynomial

complex rootsFundamental Theorem of Algebra

conjugate rootsNote

cubicExample

factoring by handParagraphs

quadraticExample

rational rootsRational Root Theorem

Power of a matrixsee Matrix multiplication

Projectionsee Orthogonal projection

diagonalizability ofExample

Projection FormulaProjection Formula

Projection matrixsee Orthogonal projection, standard matrix of

QR codesExample

Quadratic formulaExample

Rangesee Transformation

RankDefinition

rank theoremRank Theorem

row and columnSubsection

Rational Root TheoremRational Root Theorem

Real -spaceDefinition

as a subspace of itselfExample

point ofDefinition

Real numbers Paragraph Example

Real partsee Complex numbers

Red BoxExample Example Example

Reduced row echelon formDefinition

and invertibilityInvertible Matrix Theorem

Reflection

eigenspaces ofExample

eigenvectors ofExample

in generalRemark

over the -axisExample Example

inverse ofExample Example

Rotation

composition ofExample

counterclockwise by Example Example

counterclockwise by Example

eigenvectors ofExample

non-diagonalizability ofExample

Rotation-scaling matrix

and complex eigenvaluesRotation-Scaling Theorem

computing the angleImportant Note

definition ofDefinition

structure ofProposition

Rotation-Scaling TheoremRotation-Scaling Theorem

Row echelon formDefinition

Row equivalenceDefinition

Row operationsParagraphs

and determinantsDefinition

replacementItem

scalingItem

swapItem

Row ranksee Rank

Row reduction

algorithmAlgorithm

computing determinantsRecipe

picture ofParagraph

Row replacementsee Row operations, replacement

Row space

definition ofDefinition

is column space of transposeParagraph

orthogonal complement ofImportant Note

Row vectorsee Vector

Scalesee Dilationsee Row operations, scaling

Set builder notationNote

Shear

eigenvectors ofExample

in the -directionExample Example

multiplicitiesExample

non-diagonalizability ofExample

Similarity

action on a vectorImportant Note

and eigenspacesImportant Note

and eigenvaluesImportant Note

and eigenvectorsFact

and multiplicitiesTheorem

and powersFact

and the characteristic polynomialFact

and the determinantParagraph

and the traceParagraph

definition ofDefinition

equivalence relationProposition

geometry ofSubsection

identity matrixParagraph

of matricesRemark

of diagonal matricesExample

of diagonalizable matricesImportant Note

to a diagonal matrixsee Diagonalizability

worked exampleExample

Solutionsee System of linear equations

Solution set

definition ofDefinition

of a homogeneous system is a null spaceExample

of a homogeneous system is a spanImportant Note

picture ofSubsection

size ofSubsection

translate of a spanKey Observation

versus the column spaceSubsection

Space

Example

color spaceExample

dimension-3 solution setImportant Note Important Note

Span

basis ofsee Basis

definition ofDefinition

is a subspaceTheorem

orthogonal complement ofParagraph Important Note

pictures ofParagraphs

Standard coordinate vectors

and matrix columnsFact

are unit vectorsParagraph

columns of the identity matrixDefinition

definition ofNotation

orthonormality ofExample

picture ofParagraph

Standard matrixsee Linear transformation

Steady statesee Stochastic matrix

Stochastic matrix

definition ofDefinition

eigenvalues ofFact

long-term behavior ofTheorem

picture ofFigure Figure

steady state ofDefinition

computingImportant Note Important Note

sum of entries of vectorImportant Note

Subset

definition ofDefinition

set builder notationNote

versus subspaceImportant Note

Subspace

and spansTheorem

definition ofDefinition

is a spanTheorem

orthogonal complement ofsee Orthogonal complement

real -spaceExample

versus subsetImportant Note

writing as column or null spaceParagraphs

zeroExample

Superposition principleParagraph

System of linear equations

consistentDefinition

picture ofNote

span criterionNote Note

definition ofParagraph

four ways of writingNote

homogeneousDefinition

trivial solutionImportant Note

inconsistentDefinition

picture ofFigure

RREF criterionTheorem

inhomogeneousDefinition

nontrivial solutionImportant Note

and free variablesObservation

and invertibilityInvertible Matrix Theorem

findingImportant Note

number of solutions ofSubsection

parametric form ofsee Parametric form

parametric vector form ofsee Parametric vector form

particular solution ofsee Parametric vector form

solution ofDefinition

solving with the inverse matrixTheorem

trivial solutionImportant Note

and invertibilityInvertible Matrix Theorem

Tall matrixsee Matrix transformation

Tracesee Matrix

Transformation

addition ofDefinition

as a machineParagraph

associated to a matrixsee Matrix transformation

codomain ofDefinition

composition ofDefinition

noncommutativity ofImportant Note

order of operationsExample

definition ofDefinition

domain ofDefinition

identitysee Identity transformation

invertiblesee Invertible transformation

linearsee Linear transformation

of one variableExample

of several variablesExample

one-to-onesee One-to-one

ontosee Onto

range ofDefinition

scalar multiplication ofDefinition

Transposesee Matrix

Triangle

area ofsee Volume

Trivial solutionsee System of linear equations

Unit cubeExample

Unit vector

and orthonormalityDefinition

definition ofDefinition

in the direction of a vectorFact

standard coordinate vectorsParagraph

Upper-triangular

see MatrixDefinition

Vector

additionItem

parallelogram lawParagraphs

angle betweenRemark

definition ofDefinition

distance betweenDefinition

length ofFact

linear combination ofsee Linear combination

orthogonalsee Orthogonality

product with matrixsee Matrix-vector product

real and imaginary parts ofParagraph

row vectorDefinition

product with column vectorDefinition

scalar multiplicationItem

picture ofParagraphs

subtraction

picture ofParagraphs

unit vectorsee Unit vector

unit vector in the direction ofFact

Vector equation

consistentsee System of linear equations, consistent

definition ofDefinition

equivalence with matrix equationNote

equivalence with system of equationsParagraph

inconsistentsee System of linear equations, inconsistent

solvingImportant Note

Volume

and lengthExample

of a parallelepipedTheorem

of a regionTheorem

of a triangleExample

of an ellipseExample

signedRemark

Wide matrixsee Matrix transformationsee Linear independence

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