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IndexIndex

B -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 invertibilityInvertible Matrix Theorem
and orthogonal projectionCorollary
basis theoremBasis Theorem
coordinates with respect tosee B -coordinates
definition ofDefinition
infinitely manyImportant Note
making orthogonalTheorem
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 2 × 2 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 ofImportant Note
range of a transformationImportant Note
versus the solution setSubsection
Column spansee Column space
Complex conjugationsee Complex numbers
Complex eigenvalue
2 × 2 matrices
and rotation-scaling matricesRotation-Scaling Theorem
and rotation-scaling matrices, computingImportant Note
different rotation-scaling matricesParagraph
geometry ofSubsection
3 × 3 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
methods of computationImportant Note
multilinearity propertyProposition
multiplicativity propertyProposition
and volumesRemark
of a 2 × 2 matrixDefinitionExampleExample
of a 3 × 3 matrixImportant Note
of similar matricesParagraph
properties ofImportant Note
transpose propertyProposition
Diagonal
see MatrixDefinition
Diagonalizability
algebraic-geometric multiplicity criterionTheorem
criterion for similarityImportant Note
definition ofDefinition
diagonal matricesExampleExample
distinct eigenvaluesImportant Note
geometry ofSubsection
is unrelated to invertibilityImportant Note
of 2 × 2 matricesExample
of 3 × 3 matricesExample
of a projection matrixProposition
order of eigenvaluesNote
powers ofImportant Note
projectionExample
shearExample
similar matricesExample
Difference equationDefinition
eigenvectors ofExample
inverse ofExampleExample
Dimension
definition ofDefinition
of a column spaceDefinition
of a null spaceDefinition
of a solution setImportant 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 2 × 2 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 equationSubsectionNote
Inconsistentsee System of linear equations
Increasing span criterionsee Linear independence
Inhomogeneoussee System of linear equations
Invertible matrix
and invertible transformationTheorem
basic factsFact
computation
2 × 2 caseProposition
in generalTheorem
using Cramer’ ruleTheorem
definition ofDefinition
inverse ofItem
invertible matrix theoremInvertible Matrix Theorem
solving linear systems withTheorem
Invertible transformation
and invertible matricesTheoremInvertible Matrix Theorem
definition ofDefinition
one-to-one and ontoProposition
Least-squares
and Ax = b Col ( A ) Important Note
computation of
complicated matrix formulaTheorem
Projection FormulaImportant Note
row reductionTheoremImportant Note
definition ofDefinition
picture ofParagraphs
uniqueness ofTheorem
Line
dimension-1 solution setImportant Note
geometric definition ofDefinition
number lineExample
orthogonal projection ontoImportant NoteExample
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
and matrix columnsTheorem
basic factsFact
definition ofDefinition
equation of linear dependenceParagraph
increasing span criterionIncreasing Span Criterion
linear dependence relationParagraph
of an orthogonal setFact
pictures ofSubsection
redundancy criterionTheorem
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
dimensions ofImportant Note
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
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
definition ofDefinition
determinant ofProposition
dimensions ofImportant Note
inverse ofItem
noncommutativity ofExample
order of operationsExample
powersParagraph
and diagonalizabilityImportant Note
and similarityFact
properties ofNote
row-column ruleImportant 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 R 2 Example
one-to-one criteriaTheorem
onto criteriaTheorem
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 setParagraph
is a subspaceSubsection
is the 0 -eigenspaceFact
of an orthogonal projectionProposition
orthogonal complement ofImportant Note
NullityDefinition
and invertibilityInvertible Matrix Theorem
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
definition ofDefinition
dimension ofFact
of a column spaceImportant Note
of a null spaceImportant Note
of a row spaceImportant Note
of an eigenspaceExample
orthogonal complement ofFact
pictures ofParagraphs
system of linear equationsParagraph
Orthogonal decompositionsee Orthogonal projection
Orthogonal projection
and B -coordinatesImportant Note
as a transformationSubsection
composed with itselfProposition
computation of
complicated matrix formulaCorollary
Projection FormulaProjection Formula
row reductionTheoremImportant Note
definition ofDefinition
distance fromImportant Note
eigenvectors ofExample
existence ofTheorem
is the closest vectorImportant Note
linearity ofProposition
of a vector in W Example
of a vector in W Example
onto a column spaceImportant Note
onto the xy -planeExampleExampleExample
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 B -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 ofImportant Fact
zero vectorImportant Note
Orthonormal setsee Orthogonal set
Parallelepiped
definition ofDefinition
parallelogramExample
area ofExample
volume ofTheorem
Parallelogramsee Parallelepiped
Parameterized equationNote
Parametric formSubsectionImportant Note
Parametric vector form
of a homogeneous equationImportant Note
of an inhomogeneous equationParagraph
particular solutionKey Observation
Particular solutionsee Parametric vector form
Perron–Frobenius theoremTheorem
Plane
dimension-2 solution setImportant Note
geometric definition ofDefinition
parametric form ofExample
xy -planeExample
distance betweenDefinition
Polynomial
characteristicsee Characteristic polynomial
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
and invertibilityInvertible Matrix Theorem
rank theoremRank Theorem
row and columnRemark
Rational Root TheoremRational Root Theorem
Real n -spaceDefinition
as a subspace of itselfExample
point ofDefinition
Real numbers R ParagraphExample
Real partsee Complex numbers
Reduced row echelon formDefinition
and invertibilityInvertible Matrix Theorem
Reflection
eigenspaces ofExample
eigenvectors ofExample
in generalRemark
over the y -axisExampleExample
inverse ofExampleExample
Rotation
composition ofExample
counterclockwise by 45 ExampleExample
counterclockwise by 90 ExampleExample
counterclockwise by θ 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 x -directionExampleExample
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 2 × 2 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 spaceParagraph
of a homogeneous system is a spanParagraph
picture ofSubsection
size ofSubsection
translate of a spanKey Observation
versus the column spaceSubsection
Space
R 3 Example
color spaceExample
dimension-3 solution setImportant Note
Span
basis ofsee Basis
definition ofDefinition
is a subspaceTheorem
orthogonal complement ofPropositionImportant 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 ofFigureFigure
steady state ofDefinition
sum of entries of vectorImportant Note
Subset
definition ofDefinition
set builder notationNote
versus subspaceImportant Note
Subspace
definition ofDefinition
orthogonal complement ofsee Orthogonal complement
real n -spaceExample
verifyingImportant Note
versus subsetImportant Note
zeroExample
Superposition principleParagraph
System of linear equations
consistentDefinition
picture ofNote
span criterionNoteNote
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
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
Volume
and lengthExample
of a parallelepipedTheorem
of a regionTheorem
of a triangleExample
of an ellipseExample
signedRemark
Wide matrixsee Matrix transformationsee Linear independence